Sieve of Eratosthenes

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Task
Sieve of Eratosthenes
You are encouraged to solve this task according to the task description, using any language you may know.
This task has been clarified. Its programming examples are in need of review to ensure that they still fit the requirements of the task.

The Sieve of Eratosthenes is a simple algorithm that finds the prime numbers up to a given integer. Implement this algorithm, with the only allowed optimization that the outer loop can stop at the square root of the limit, and the inner loop may start at the square of the prime just found. That means especially that you shouldn't optimize by using pre-computed wheels, i.e. don't assume you need only to cross out odd numbers (wheel based on 2), numbers equal to 1 or 5 modulo 6 (wheel based on 2 and 3), or similar wheels based on low primes.

If there's an easy way to add such a wheel based optimization, implement this as an alternative version.

Note
  • It is important that the sieve algorithm be the actual algorithm used to find prime numbers for the task.
Cf

Contents

[edit] 360 Assembly

For maximum compatibility, this program uses only the basic instruction set.

*        Sieve of Eratosthenes 
ERATOST CSECT
USING ERATOST,R12
SAVEAREA B STM-SAVEAREA(R15)
DC 17F'0'
DC CL8'ERATOST'
STM STM R14,R12,12(R13) save calling context
ST R13,4(R15)
ST R15,8(R13)
LR R12,R15 set addessability
* ---- CODE
LA R4,1 I=1
LA R6,1 increment
L R7,N limit
LOOPI BXH R4,R6,ENDLOOPI do I=2 to N
LR R1,R4 R1=I
BCTR R1,0
LA R14,CRIBLE(R1)
CLI 0(R14),X'01'
BNE ENDIF if not CRIBLE(I)
LR R5,R4 J=I
LR R8,R4
LR R9,R7
LOOPJ BXH R5,R8,ENDLOOPJ do J=I*2 to N by I
LR R1,R5 R1=J
BCTR R1,0
LA R14,CRIBLE(R1)
MVI 0(R14),X'00' CRIBLE(J)='0'B
B LOOPJ
ENDLOOPJ EQU *
ENDIF EQU *
B LOOPI
ENDLOOPI EQU *
LA R4,1 I=1
LA R6,1
L R7,N
LOOP BXH R4,R6,ENDLOOP do I=1 to N
LR R1,R4 R1=I
BCTR R1,0
LA R14,CRIBLE(R1)
CLI 0(R14),X'01'
BNE NOTPRIME if not CRIBLE(I)
CVD R4,P P=I
UNPK Z,P Z=P
MVC C,Z C=Z
OI C+L'C-1,X'F0' zap sign
MVC WTOBUF(8),C+8
WTO MF=(E,WTOMSG)
NOTPRIME EQU *
B LOOP
ENDLOOP EQU *
RETURN EQU *
LM R14,R12,12(R13) restore context
XR R15,R15 set return code to 0
BR R14 return to caller
* ---- DATA
I DS F
J DS F
DS 0F
P DS PL8 packed
Z DS ZL16 zoned
C DS CL16 character
WTOMSG CNOP 0,4
DC H'80' length of WTO buffer
DC H'0' must be binary zeroes
WTOBUF DC 80C' '
LTORG
N DC F'100000'
CRIBLE DC 100000X'01'
YREGS
END ERATOST
Output:
00000002
00000003
00000005
00000007
00000011
00000013
00000017
00000019
00000023
00000029
00000031
00000037
00000041
00000043
00000047
00000053
00000059
00000061
00000067
...
00099767
00099787
00099793
00099809
00099817
00099823
00099829
00099833
00099839
00099859
00099871
00099877
00099881
00099901
00099907
00099923
00099929
00099961
00099971
00099989
00099991

[edit] 68000 Assembly

Algorithm somewhat optimized: array omits 1, 2, all higher odd numbers. Optimized for storage: uses bit array for prime/composite flags.

Works with: [EASy68K v5.13.00]

Some of the macro code is derived from the examples included with EASy68K. See 68000 "100 Doors" listing for additional information.

*-----------------------------------------------------------
* Title  : BitSieve
* Written by : G. A. Tippery
* Date  : 2014-Feb-24, 2013-Dec-22
* Description: Prime number sieve
*-----------------------------------------------------------
ORG $1000
 
** ---- Generic macros ---- **
PUSH MACRO
MOVE.L \1,-(SP)
ENDM
 
POP MACRO
MOVE.L (SP)+,\1
ENDM
 
DROP MACRO
ADDQ #4,SP
ENDM
 
PUTS MACRO
** Print a null-terminated string w/o CRLF **
** Usage: PUTS stringaddress
** Returns with D0, A1 modified
MOVEQ #14,D0 ; task number 14 (display null string)
LEA \1,A1 ; address of string
TRAP #15 ; display it
ENDM
 
GETN MACRO
MOVEQ #4,D0 ; Read a number from the keyboard into D1.L.
TRAP #15
ENDM
 
** ---- Application-specific macros ---- **
 
val MACRO ; Used by bit sieve. Converts bit address to the number it represents.
ADD.L \1,\1 ; double it because odd numbers are omitted
ADDQ #3,\1 ; add offset because initial primes (1, 2) are omitted
ENDM
 
* ** ================================================================================ **
* ** Integer square root routine, bisection method **
* ** IN: D0, should be 0<D0<$1000 (65536) -- higher values MAY work, no guarantee
* ** OUT: D1
*
SquareRoot:
*
MOVEM.L D2-D4,-(SP) ; save registers needed for local variables
* DO == n
* D1 == a
* D2 == b
* D3 == guess
* D4 == temp
*
* a = 1;
* b = n;
MOVEQ #1,D1
MOVE.L D0,D2
* do {
REPEAT
* guess = (a+b)/2;
MOVE.L D1,D3
ADD.L D2,D3
LSR.L #1,D3
* if (guess*guess > n) { // inverse function of sqrt is square
MOVE.L D3,D4
MULU D4,D4 ; guess^2
CMP.L D0,D4
BLS .else
* b = guess;
MOVE.L D3,D2
BRA .endif
* } else {
.else:
* a = guess;
MOVE.L D3,D1
* } //if
.endif:
* } while ((b-a) > 1); ; Same as until (b-a)<=1 or until (a-b)>=1
MOVE.L D2,D4
SUB.L D1,D4 ; b-a
UNTIL.L D4 <LE> #1 DO.S
* return (a) ; Result is in D1
* } //LongSqrt()
MOVEM.L (SP)+,D2-D4 ; restore saved registers
RTS
*
* ** ================================================================================ **
 
 
** ======================================================================= **
*
** Prime-number Sieve of Eratosthenes routine using a big bit field for flags **
* Enter with D0 = size of sieve (bit array)
* Prints found primes 10 per line
* Returns # prime found in D6
*
* Register usage:
*
* D0 == n
* D1 == prime
* D2 == sqroot
* D3 == PIndex
* D4 == CIndex
* D5 == MaxIndex
* D6 == PCount
*
* A0 == PMtx[0]
*
* On return, all registers above except D0 are modified. Could add MOVEMs to save and restore D2-D6/A0.
*
 
** ------------------------ **
 
GetBit: ** sub-part of Sieve subroutine **
** Entry: bit # is on TOS
** Exit: A6 holds the byte number, D7 holds the bit number within the byte
** Note: Input param is still on TOS after return. Could have passed via a register, but
** wanted to practice with stack. :)
*
MOVE.L (4,SP),D7 ; get value from (pre-call) TOS
ASR.L #3,D7 ; /8
MOVEA D7,A6 ; byte #
MOVE.L (4,SP),D7 ; get value from (pre-call) TOS
AND.L #$7,D7 ; bit #
RTS
 
** ------------------------ **
 
Sieve:
MOVE D0,D5
SUBQ #1,D5
JSR SquareRoot ; sqrt D0 => D1
MOVE.L D1,D2
LEA PArray,A0
CLR.L D3
*
PrimeLoop:
MOVE.L D3,D1
val D1
MOVE.L D3,D4
ADD.L D1,D4
*
CxLoop: ; Goes through array marking multiples of d1 as composite numbers
CMP.L D5,D4
BHI ExitCx
PUSH D4 ; set D7 as bit # and A6 as byte pointer for D4'th bit of array
JSR GetBit
DROP
BSET D7,0(A0,A6.L) ; set bit to mark as composite number
ADD.L D1,D4 ; next number to mark
BRA CxLoop
ExitCx:
CLR.L D1 ; Clear new-prime-found flag
ADDQ #1,D3 ; Start just past last prime found
PxLoop: ; Searches for next unmarked (not composite) number
CMP.L D2,D3 ; no point searching past where first unmarked multiple would be past end of array
BHI ExitPx ; if past end of array
TST.L D1
BNE ExitPx ; if flag set, new prime found
PUSH D3 ; check D3'th bit flag
JSR GetBit ; sets D7 as bit # and A6 as byte pointer
DROP ; drop TOS
BTST D7,0(A0,A6.L) ; read bit flag
BNE IsSet ; If already tagged as composite
MOVEQ #-1,D1 ; Set flag that we've found a new prime
IsSet:
ADDQ #1,D3 ; next PIndex
BRA PxLoop
ExitPx:
SUBQ #1,D3 ; back up PIndex
TST.L D1 ; Did we find a new prime #?
BNE PrimeLoop ; If another prime # found, go process it
*
; fall through to print routine
 
** ------------------------ **
 
* Print primes found
*
* D4 == Column count
*
* Print header and assumed primes (#1, #2)
PUTS Header ; Print string @ Header, no CR/LF
MOVEQ #2,D6 ; Start counter at 2 because #1 and #2 are assumed primes
MOVEQ #2,D4
*
MOVEQ #0,D3
PrintLoop:
CMP.L D5,D3
BHS ExitPL
PUSH D3
JSR GetBit ; sets D7 as bit # and A6 as byte pointer
DROP ; drop TOS
BTST D7,0(A0,A6.L)
BNE NotPrime
* printf(" %6d", val(PIndex)
MOVE.L D3,D1
val D1
AND.L #$0000FFFF,D1
MOVEQ #6,D2
MOVEQ #20,D0 ; display signed RJ
TRAP #15
ADDQ #1,D4
ADDQ #1,D6
* *** Display formatting ***
* if((PCount % 10) == 0) printf("\n");
CMP #10,D4
BLO NoLF
PUTS CRLF
MOVEQ #0,D4
NoLF:
NotPrime:
ADDQ #1,D3
BRA PrintLoop
ExitPL:
RTS
 
** ======================================================================= **
 
N EQU 5000 ; *** Size of boolean (bit) array ***
SizeInBytes EQU (N+7)/8
*
START: ; first instruction of program
MOVE.L #N,D0 ; # to test
JSR Sieve
* printf("\n %d prime numbers found.\n", D6); ***
PUTS Summary1,A1
MOVE #3,D0 ; Display signed number in D1.L in decimal in smallest field.
MOVE.W D6,D1
TRAP #15
PUTS Summary2,A1
 
SIMHALT ; halt simulator
 
** ======================================================================= **
 
* Variables and constants here
 
ORG $2000
CR EQU 13
LF EQU 10
CRLF DC.B CR,LF,$00
 
PArray: DCB.B SizeInBytes,0
 
Header: DC.B CR,LF,LF,' Primes',CR,LF,' ======',CR,LF
DC.B ' 1 2',$00
 
Summary1: DC.B CR,LF,' ',$00
Summary2: DC.B ' prime numbers found.',CR,LF,$00
 
END START ; last line of source

[edit] ACL2

(defun nats-to-from (n i)
(declare (xargs :measure (nfix (- n i))))
(if (zp (- n i))
nil
(cons i (nats-to-from n (+ i 1)))))
 
(defun remove-multiples-up-to-r (factor limit xs i)
(declare (xargs :measure (nfix (- limit i))))
(if (or (> i limit)
(zp (- limit i))
(zp factor))
xs
(remove-multiples-up-to-r
factor
limit
(remove i xs)
(+ i factor))))
 
(defun remove-multiples-up-to (factor limit xs)
(remove-multiples-up-to-r factor limit xs (* factor 2)))
 
(defun sieve-r (factor limit)
(declare (xargs :measure (nfix (- limit factor))))
(if (zp (- limit factor))
(nats-to-from limit 2)
(remove-multiples-up-to factor (+ limit 1)
(sieve-r (1+ factor) limit))))
 
(defun sieve (limit)
(sieve-r 2 limit))

[edit] Ada

with Ada.Text_IO, Ada.Command_Line;
 
procedure Eratos is
 
Last: Positive := Positive'Value(Ada.Command_Line.Argument(1));
Prime: array(1 .. Last) of Boolean := (1 => False, others => True);
Base: Positive := 2;
Cnt: Positive;
begin
loop
exit when Base * Base > Last;
if Prime(Base) then
Cnt := Base + Base;
loop
exit when Cnt > Last;
Prime(Cnt) := False;
Cnt := Cnt + Base;
end loop;
end if;
Base := Base + 1;
end loop;
Ada.Text_IO.Put("Primes less or equal" & Positive'Image(Last) &" are:");
for Number in Prime'Range loop
if Prime(Number) then
Ada.Text_IO.Put(Positive'Image(Number));
end if;
end loop;
end Eratos;
Output:
> ./eratos 31
Primes less or equal 31 are : 2 3 5 7 11 13 17 19 23 29 31

[edit] ALGOL 60

Works with: ALGOL 60 for OS/360

'BEGIN'
'INTEGER' 'ARRAY' CANDIDATES(/0..1000/);
'INTEGER' I,J,K;
'COMMENT' SET LINE-LENGTH=120,SET LINES-PER-PAGE=62,OPEN;
SYSACT(1,6,120); SYSACT(1,8,62); SYSACT(1,12,1);
'FOR' I := 0 'STEP' 1 'UNTIL' 1000 'DO'
'BEGIN'
CANDIDATES(/I/) := 1;
'END';
CANDIDATES(/0/) := 0;
CANDIDATES(/1/) := 0;
I := 0;
'FOR' I := I 'WHILE' I 'LESS' 1000 'DO'
'BEGIN'
'FOR' I := I 'WHILE' I 'LESS' 1000
'AND' CANDIDATES(/I/) 'EQUAL' 0 'DO'
I := I+1;
'IF' I 'LESS' 1000 'THEN'
'BEGIN'
J := 2;
K := J*I;
'FOR' K := K 'WHILE' K 'LESS' 1000 'DO'
'BEGIN'
CANDIDATES(/K/) := 0;
J := J + 1;
K := J*I;
'END';
I := I+1;
'END'
'END';
'FOR' I := 0 'STEP' 1 'UNTIL' 999 'DO'
'IF' CANDIDATES(/I/) 'NOTEQUAL' 0 'THEN'
'BEGIN'
OUTINTEGER(1,I);
OUTSTRING(1,'(' IS PRIME')');
'COMMENT' NEW LINE;
SYSACT(1,14,1)
'END'
'END'
'END'

[edit] ALGOL 68

BOOL prime = TRUE, non prime = FALSE;
PROC eratosthenes = (INT n)[]BOOL:
(
[n]BOOL sieve;
FOR i TO UPB sieve DO sieve[i] := prime OD;
INT m = ENTIER sqrt(n);
sieve[1] := non prime;
FOR i FROM 2 TO m DO
IF sieve[i] = prime THEN
FOR j FROM i*i BY i TO n DO
sieve[j] := non prime
OD
FI
OD;
sieve
);
 
print((eratosthenes(80),new line))
Output:
FTTFTFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFFFFFTFFFFFTFTFFFFFTFFFTFTFFFFFTF

[edit] AutoHotkey

Search autohotkey.com: of Eratosthenes
Source: AutoHotkey forum by Laszlo

MsgBox % "12345678901234567890`n" Sieve(20) 
 
Sieve(n) { ; Sieve of Eratosthenes => string of 0|1 chars, 1 at position k: k is prime
Static zero := 48, one := 49 ; Asc("0"), Asc("1")
VarSetCapacity(S,n,one)
NumPut(zero,S,0,"char")
i := 2
Loop % sqrt(n)-1 {
If (NumGet(S,i-1,"char") = one)
Loop % n//i
If (A_Index > 1)
NumPut(zero,S,A_Index*i-1,"char")
i += 1+(i>2)
}
Return S
}

[edit] AutoIt

#include <Array.au3>
$M = InputBox("Integer", "Enter biggest Integer")
Global $a[$M], $r[$M], $c = 1
For $i = 2 To $M -1
If Not $a[$i] Then
$r[$c] = $i
$c += 1
For $k = $i To $M -1 Step $i
$a[$k] = True
Next
EndIf
Next
$r[0] = $c - 1
ReDim $r[$c]
_ArrayDisplay($r)

[edit] AWK

An initial array holds all numbers 2..max (which is entered on stdin); then all products of integers are deleted from it; the remaining are displayed in the unsorted appearance of a hash table. Here, the script is entered directly on the commandline, and input entered on stdin:

$ awk '{for(i=2;i<=$1;i++) a[i]=1;
>       for(i=2;i<=sqrt($1);i++) for(j=2;j<=$1;j++) delete a[i*j];
>       for(i in a) printf i" "}'
100
71 53 17 5 73 37 19 83 47 29 7 67 59 11 97 79 89 31 13 41 23 2 61 43 3

The following variant does not unset non-primes, but sets them to 0, to preserve order in output:

$ awk '{for(i=2;i<=$1;i++) a[i]=1;
>       for(i=2;i<=sqrt($1);i++) for(j=2;j<=$1;j++) a[i*j]=0;
>       for(i=2;i<=$1;i++) if(a[i])printf i" "}'
100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

Now with the script from a file, input from commandline as well as stdin, and input is checked for valid numbers:

 
# usage: gawk -v n=101 -f sieve.awk
 
function sieve(n) { # print n,":"
for(i=2; i<=n; i++) a[i]=1;
for(i=2; i<=sqrt(n);i++) for(j=2;j<=n;j++) a[i*j]=0;
for(i=2; i<=n; i++) if(a[i]) printf i" "
print ""
}
 
BEGIN { print "Sieve of Eratosthenes:"
if(n>1) sieve(n)
}
 
{ n=$1+0 }
n<2 { exit }
{ sieve(n) }
 
END { print "Bye!" }
 

[edit] BASIC

Works with: FreeBASIC
Works with: RapidQ
DIM n AS Integer, k AS Integer, limit AS Integer
 
INPUT "Enter number to search to: "; limit
DIM flags(limit) AS Integer
 
FOR n = 2 TO SQR(limit)
IF flags(n) = 0 THEN
FOR k = n*n TO limit STEP n
flags(k) = 1
NEXT k
END IF
NEXT n
 
' Display the primes
FOR n = 2 TO limit
IF flags(n) = 0 THEN PRINT n; ", ";
NEXT n

[edit] Applesoft BASIC

10  INPUT "ENTER NUMBER TO SEARCH TO: ";LIMIT
20 DIM FLAGS(LIMIT)
30 FOR N = 2 TO SQR (LIMIT)
40 IF FLAGS(N) < > 0 GOTO 80
50 FOR K = N * N TO LIMIT STEP N
60 FLAGS(K) = 1
70 NEXT K
80 NEXT N
90 REM DISPLAY THE PRIMES
100 FOR N = 2 TO LIMIT
110 IF FLAGS(N) = 0 THEN PRINT N;", ";
120 NEXT N

[edit] Locomotive Basic

10 DEFINT a-z
20 INPUT "Limit";limit
30 DIM f(limit)
40 FOR n=2 TO SQR(limit)
50 IF f(n)=1 THEN 90
60 FOR k=n*n TO limit STEP n
70 f(k)=1
80 NEXT k
90 NEXT n
100 FOR n=2 TO limit
110 IF f(n)=0 THEN PRINT n;",";
120 NEXT

[edit] ZX Spectrum Basic

10 INPUT "Enter number to search to: ";l
20 DIM p(l)
30 FOR n=2 TO SQR l
40 IF p(n)<>0 THEN NEXT n
50 FOR k=n*n TO l STEP n
60 LET p(k)=1
70 NEXT k
80 NEXT n
90 REM Display the primes
100 FOR n=2 TO l
110 IF p(n)=0 THEN PRINT n;", ";
120 NEXT n

[edit] BBC BASIC

      limit% = 100000
DIM sieve% limit%
 
prime% = 2
WHILE prime%^2 < limit%
FOR I% = prime%*2 TO limit% STEP prime%
sieve%?I% = 1
NEXT
REPEAT prime% += 1 : UNTIL sieve%?prime%=0
ENDWHILE
 
REM Display the primes:
FOR I% = 1 TO limit%
IF sieve%?I% = 0 PRINT I%;
NEXT

[edit] bash

See solutions at UNIX Shell.

[edit] Batch File

:: Sieve of Eratosthenes for Rosetta Code - PG
@echo off
setlocal ENABLEDELAYEDEXPANSION
setlocal ENABLEEXTENSIONS
rem echo on
set /p n=limit:
rem set n=100
for /L %%i in (1,1,%n%) do set crible.%%i=1
for /L %%i in (2,1,%n%) do (
if !crible.%%i! EQU 1 (
set /A w = %%i * 2
for /L %%j in (!w!,%%i,%n%) do (
set crible.%%j=0
)
)
)
for /L %%i in (2,1,%n%) do (
if !crible.%%i! EQU 1 echo %%i
)
pause
Output:
limit: 100
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97

[edit] Befunge

2>:3g" "-!v\  g30          <
 |!`"O":+1_:.:03p>03g+:"O"`|
 @               ^  p3\" ":<
2 234567890123456789012345678901234567890123456789012345678901234567890123456789

[edit] Bracmat

This solution does not use an array. Instead, numbers themselves are used as variables. The numbers that are not prime are set (to the silly value "nonprime"). Finally all numbers up to the limit are tested for being initialised. The uninitialised (unset) ones must be the primes.

( ( eratosthenes
= n j i
.  !arg:?n
& 1:?i
& whl
' ( (1+!i:?i)^2:~>!n:?j
& ( !!i
| whl
' ( !j:~>!n
& nonprime:?!j
& !j+!i:?j
)
)
)
& 1:?i
& whl
' ( 1+!i:~>!n:?i
& (!!i|put$(!i " "))
)
)
& eratosthenes$100
)
Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

[edit] C

Plain sieve, without any optimizations:
#include <stdlib.h>
#include <math.h>
 
char*
eratosthenes(int n, int *c)
{
char* sieve;
int i, j, m;
 
if(n < 2)
return NULL;
 
*c = n-1; /* primes count */
m = (int) sqrt((double) n);
 
/* calloc initializes to zero */
sieve = calloc(n+1,sizeof(char));
sieve[0] = 1;
sieve[1] = 1;
for(i = 2; i <= m; i++)
if(!sieve[i])
for (j = i*i; j <= n; j += i)
if(!sieve[j]){
sieve[j] = 1;
--(*c);
}
return sieve;
}
Possible optimizations include sieving only odd numbers (or more complex wheels), packing the sieve into bits to improve locality (and allow larger sieves), etc.

Another example:

We first fill ones into an array and assume all numbers are prime. Then, in a loop, fill zeroes into those places where i * j is less than or equal to n (number of primes requested), which means they have multiples! To understand this better, look at the output of the following example.

To print this back, we look for ones in the array and only print those spots.
#include <stdio.h>
#include <malloc.h>
void sieve(int *, int);
 
int main(int argc, char *argv)
{
int *array, n=10;
array =(int *)malloc(sizeof(int));
sieve(array,n);
return 0;
}
 
void sieve(int *a, int n)
{
int i=0, j=0;
 
for(i=2; i<=n; i++) {
a[i] = 1;
}
 
for(i=2; i<=n; i++) {
printf("\ni:%d", i);
if(a[i] == 1) {
for(j=i; (i*j)<=n; j++) {
printf ("\nj:%d", j);
printf("\nBefore a[%d*%d]: %d", i, j, a[i*j]);
a[(i*j)] = 0;
printf("\nAfter a[%d*%d]: %d", i, j, a[i*j]);
}
}
}
 
printf("\nPrimes numbers from 1 to %d are : ", n);
for(i=2; i<=n; i++) {
if(a[i] == 1)
printf("%d, ", i);
}
printf("\n\n");
}
Output:
i:2
j:2
Before a[2*2]: 1
After a[2*2]: 0
j:3
Before a[2*3]: 1
After a[2*3]: 0
j:4
Before a[2*4]: 1
After a[2*4]: 0
j:5
Before a[2*5]: 1
After a[2*5]: 0
i:3
j:3
Before a[3*3]: 1
After a[3*3]: 0
i:4
i:5
i:6
i:7
i:8
i:9
i:10
Primes numbers from 1 to 10 are : 2, 3, 5, 7,

[edit] C++

// yield all prime numbers less than limit. 
template<class UnaryFunction>
void primesupto(int limit, UnaryFunction yield)
{
std::vector<bool> is_prime(limit, true);
 
const int sqrt_limit = static_cast<int>(std::sqrt(limit));
for (int n = 2; n <= sqrt_limit; ++n)
if (is_prime[n]) {
yield(n);
 
for (unsigned k = n*n, ulim = static_cast<unsigned>(limit); k < ulim; k += n)
//NOTE: "unsigned" is used to avoid an overflow in `k+=n` for `limit` near INT_MAX
is_prime[k] = false;
}
 
for (int n = sqrt_limit + 1; n < limit; ++n)
if (is_prime[n])
yield(n);
}

Full program:

Works with: Boost
/**
$ g++ -I/path/to/boost sieve.cpp -o sieve && sieve 10000000
*/

#include <inttypes.h> // uintmax_t
#include <limits>
#include <cmath>
#include <iostream>
#include <sstream>
#include <vector>
 
#include <boost/lambda/lambda.hpp>
 
 
int main(int argc, char *argv[])
{
using namespace std;
using namespace boost::lambda;
 
int limit = 10000;
if (argc == 2) {
stringstream ss(argv[--argc]);
ss >> limit;
 
if (limit < 1 or ss.fail()) {
cerr << "USAGE:\n sieve LIMIT\n\nwhere LIMIT in the range [1, "
<< numeric_limits<int>::max() << ")" << endl;
return 2;
}
}
 
// print primes less then 100
primesupto(100, cout << _1 << " ");
cout << endl;
 
// find number of primes less then limit and their sum
int count = 0;
uintmax_t sum = 0;
primesupto(limit, (var(sum) += _1, var(count) += 1));
 
cout << "limit sum pi(n)\n"
<< limit << " " << sum << " " << count << endl;
}

[edit] C#

Works with: C# version 2.0+
using System;
using System.Collections;
using System.Collections.Generic;
 
namespace SieveOfEratosthenes
{
class Program
{
static void Main(string[] args)
{
int maxprime = int.Parse(args[0]);
var primelist = GetAllPrimesLessThan(maxprime);
foreach (int prime in primelist)
{
Console.WriteLine(prime);
}
Console.WriteLine("Count = " + primelist.Count);
Console.ReadLine();
}
 
private static List<int> GetAllPrimesLessThan(int maxPrime)
{
var primes = new List<int>() { 2 };
var maxSquareRoot = Math.Sqrt(maxPrime);
var eliminated = new BitArray(maxPrime + 1);
 
for (int i = 3; i <= maxPrime; i += 2)
{
if (!eliminated[i])
{
primes.Add(i);
if (i < maxSquareRoot)
{
for (int j = i * i; j <= maxPrime; j += 2 * i)
{
eliminated[j] = true;
}
}
}
}
return primes;
}
}
}

[edit] Unbounded

Richard Bird Sieve

Translation of: F#

To show that C# code can be written in somewhat functional paradigms, the following in an implementation of the Richard Bird sieve from the Epilogue of Melissa E. O'Neill's definitive article in Haskell:

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = UInt32;
class PrimesBird : IEnumerable<PrimeT> {
private struct CIS<T> {
public T v; public Func<CIS<T>> cont;
public CIS(T v, Func<CIS<T>> cont) {
this.v = v; this.cont = cont;
}
}
private CIS<PrimeT> pmlts(PrimeT p) {
Func<PrimeT, CIS<PrimeT>> fn = null;
fn = (c) => new CIS<PrimeT>(c, () => fn(c + p));
return fn(p * p);
}
private CIS<CIS<PrimeT>> allmlts(CIS<PrimeT> ps) {
return new CIS<CIS<PrimeT>>(pmlts(ps.v), () => allmlts(ps.cont())); }
private CIS<PrimeT> merge(CIS<PrimeT> xs, CIS<PrimeT> ys) {
var x = xs.v; var y = ys.v;
if (x < y) return new CIS<PrimeT>(x, () => merge(xs.cont(), ys));
else if (y < x) return new CIS<PrimeT>(y, () => merge(xs, ys.cont()));
else return new CIS<PrimeT>(x, () => merge(xs.cont(), ys.cont()));
}
private CIS<PrimeT> cmpsts(CIS<CIS<PrimeT>> css) {
return new CIS<PrimeT>(css.v.v, () => merge(css.v.cont(), cmpsts(css.cont()))); }
private CIS<PrimeT> minusat(PrimeT n, CIS<PrimeT> cs) {
var nn = n; var ncs = cs;
for (; ; ++nn) {
if (nn >= ncs.v) ncs = ncs.cont();
else return new CIS<PrimeT>(nn, () => minusat(++nn, ncs));
}
}
private CIS<PrimeT> prms() {
return new CIS<PrimeT>(2, () => minusat(3, cmpsts(allmlts(prms())))); }
public IEnumerator<PrimeT> GetEnumerator() {
for (var ps = prms(); ; ps = ps.cont()) yield return ps.v;
}
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}

Tree Folding Sieve

Translation of: F#

The above code can easily be converted to "odds-only" and a infinite tree-like folding scheme with the following minor changes:

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = UInt32;
class PrimesTreeFold : IEnumerable<PrimeT> {
private struct CIS<T> {
public T v; public Func<CIS<T>> cont;
public CIS(T v, Func<CIS<T>> cont) {
this.v = v; this.cont = cont;
}
}
private CIS<PrimeT> pmlts(PrimeT p) {
var adv = p + p;
Func<PrimeT, CIS<PrimeT>> fn = null;
fn = (c) => new CIS<PrimeT>(c, () => fn(c + adv));
return fn(p * p);
}
private CIS<CIS<PrimeT>> allmlts(CIS<PrimeT> ps) {
return new CIS<CIS<PrimeT>>(pmlts(ps.v), () => allmlts(ps.cont()));
}
private CIS<PrimeT> merge(CIS<PrimeT> xs, CIS<PrimeT> ys) {
var x = xs.v; var y = ys.v;
if (x < y) return new CIS<PrimeT>(x, () => merge(xs.cont(), ys));
else if (y < x) return new CIS<PrimeT>(y, () => merge(xs, ys.cont()));
else return new CIS<PrimeT>(x, () => merge(xs.cont(), ys.cont()));
}
private CIS<CIS<PrimeT>> pairs(CIS<CIS<PrimeT>> css) {
var nxtcss = css.cont();
return new CIS<CIS<PrimeT>>(merge(css.v, nxtcss.v), () => pairs(nxtcss.cont())); }
private CIS<PrimeT> cmpsts(CIS<CIS<PrimeT>> css) {
return new CIS<PrimeT>(css.v.v, () => merge(css.v.cont(), cmpsts(pairs(css.cont()))));
}
private CIS<PrimeT> minusat(PrimeT n, CIS<PrimeT> cs) {
var nn = n; var ncs = cs;
for (; ; nn += 2) {
if (nn >= ncs.v) ncs = ncs.cont();
else return new CIS<PrimeT>(nn, () => minusat(nn + 2, ncs));
}
}
private CIS<PrimeT> oddprms() {
return new CIS<PrimeT>(3, () => minusat(5, cmpsts(allmlts(oddprms()))));
}
public IEnumerator<PrimeT> GetEnumerator() {
yield return 2;
for (var ps = oddprms(); ; ps = ps.cont()) yield return ps.v;
}
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}

The above code runs over ten times faster than the original Richard Bird algorithm.

Priority Queue Sieve

Translation of: F#

First, an implementation of a Min Heap Priority Queue is provided as extracted from the entry at RosettaCode, with only the necessary methods duplicated here:

namespace PriorityQ {
using KeyT = UInt32;
using System;
using System.Collections.Generic;
using System.Linq;
class Tuple<K, V> { // for DotNet 3.5 without Tuple's
public K Item1; public V Item2;
public Tuple(K k, V v) { Item1 = k; Item2 = v; }
public override string ToString() {
return "(" + Item1.ToString() + ", " + Item2.ToString() + ")";
}
}
class MinHeapPQ<V> {
private struct HeapEntry {
public KeyT k; public V v;
public HeapEntry(KeyT k, V v) { this.k = k; this.v = v; }
}
private List<HeapEntry> pq;
private MinHeapPQ() { this.pq = new List<HeapEntry>(); }
private bool mt { get { return pq.Count == 0; } }
private Tuple<KeyT, V> pkmn {
get {
if (pq.Count == 0) return null;
else {
var mn = pq[0];
return new Tuple<KeyT, V>(mn.k, mn.v);
}
}
}
private void psh(KeyT k, V v) { // add extra very high item if none
if (pq.Count == 0) pq.Add(new HeapEntry(UInt32.MaxValue, v));
var i = pq.Count; pq.Add(pq[i - 1]); // copy bottom item...
for (var ni = i >> 1; ni > 0; i >>= 1, ni >>= 1) {
var t = pq[ni - 1];
if (t.k > k) pq[i - 1] = t; else break;
}
pq[i - 1] = new HeapEntry(k, v);
}
private void siftdown(KeyT k, V v, int ndx) {
var cnt = pq.Count - 1; var i = ndx;
for (var ni = i + i + 1; ni < cnt; ni = ni + ni + 1) {
var oi = i; var lk = pq[ni].k; var rk = pq[ni + 1].k;
var nk = k;
if (k > lk) { i = ni; nk = lk; }
if (nk > rk) { ni += 1; i = ni; }
if (i != oi) pq[oi] = pq[i]; else break;
}
pq[i] = new HeapEntry(k, v);
}
private void rplcmin(KeyT k, V v) {
if (pq.Count > 1) siftdown(k, v, 0); }
public static MinHeapPQ<V> empty { get { return new MinHeapPQ<V>(); } }
public static Tuple<KeyT, V> peekMin(MinHeapPQ<V> pq) { return pq.pkmn; }
public static MinHeapPQ<V> push(KeyT k, V v, MinHeapPQ<V> pq) {
pq.psh(k, v); return pq; }
public static MinHeapPQ<V> replaceMin(KeyT k, V v, MinHeapPQ<V> pq) {
pq.rplcmin(k, v); return pq; }
}

The following code implements an improved version of the odds-only O'Neil algorithm, which provides the improvements of only adding base prime composite number streams to the queue when the sieved number reaches the square of the base prime (saving a huge amount of memory and considerable execution time, including not needing as wide a range of a type for the internal prime numbers) as well as minimizing stream processing using fusion:

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = UInt32;
class PrimesPQ : IEnumerable<PrimeT> {
private IEnumerator<PrimeT> nmrtr() {
MinHeapPQ<PrimeT> pq = MinHeapPQ<PrimeT>.empty;
PrimeT bp = 3; PrimeT q = 9;
IEnumerator<PrimeT> bps = null;
yield return 2; yield return 3;
for (var n = (PrimeT)5; ; n += 2) {
if (n >= q) { // always equal or less...
if (q <= 9) {
bps = nmrtr();
bps.MoveNext(); bps.MoveNext(); } // move to 3...
bps.MoveNext(); var nbp = bps.Current; q = nbp * nbp;
var adv = bp + bp; bp = nbp;
pq = MinHeapPQ<PrimeT>.push(n + adv, adv, pq);
}
else {
var pk = MinHeapPQ<PrimeT>.peekMin(pq);
var ck = (pk == null) ? q : pk.Item1;
if (n >= ck) {
do { var adv = pk.Item2;
pq = MinHeapPQ<PrimeT>.replaceMin(ck + adv, adv, pq);
pk = MinHeapPQ<PrimeT>.peekMin(pq); ck = pk.Item1;
} while (n >= ck);
}
else yield return n;
}
}
}
public IEnumerator<PrimeT> GetEnumerator() { return nmrtr(); }
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}

The above code is at least about 2.5 times faster than the Tree Folding version.

Dictionary (Hash table) Sieve

The above code adds quite a bit of overhead in having to provide a version of a Priority Queue for little advantage over a Dictionary (hash table based) version as per the code below:

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = UInt32;
class PrimesDict : IEnumerable<PrimeT> {
private IEnumerator<PrimeT> nmrtr() {
Dictionary<PrimeT, PrimeT> dct = new Dictionary<PrimeT, PrimeT>();
PrimeT bp = 3; PrimeT q = 9;
IEnumerator<PrimeT> bps = null;
yield return 2; yield return 3;
for (var n = (PrimeT)5; ; n += 2) {
if (n >= q) { // always equal or less...
if (q <= 9) {
bps = nmrtr();
bps.MoveNext(); bps.MoveNext();
} // move to 3...
bps.MoveNext(); var nbp = bps.Current; q = nbp * nbp;
var adv = bp + bp; bp = nbp;
dct.Add(n + adv, adv);
}
else {
if (dct.ContainsKey(n)) {
PrimeT nadv; dct.TryGetValue(n, out nadv); dct.Remove(n); var nc = n + nadv;
while (dct.ContainsKey(nc)) nc += nadv;
dct.Add(nc, nadv);
}
else yield return n;
}
}
}
public IEnumerator<PrimeT> GetEnumerator() { return nmrtr(); }
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}

The above code runs in about three quarters of the time as the above Priority Queue based version for a range of a million primes which will fall even further behind for increasing ranges due to the Dictionary providing O(1) access times as compared to the O(log n) access times for the Priority Queue; the only slight advantage of the PQ based version is at very small ranges where the constant factor overhead of computing the table hashes becomes greater than the "log n" factor for small "n".

Page Segmented Array Sieve

All of the above unbounded versions are really just an intellectual exercise as with very little extra lines of code above the fastest Dictionary based version, one can have an bit-packed page-segmented array based version as follows:

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = UInt32;
class PrimesPgd : IEnumerable<PrimeT> {
private const int PGSZ = 1 << 14; // L1 CPU cache size in bytes
private const int BFBTS = PGSZ * 8; // in bits
private const int BFRNG = BFBTS * 2;
public IEnumerator<PrimeT> nmrtr() {
IEnumerator<PrimeT> bps = null;
List<uint> bpa = new List<uint>();
uint[] cbuf = new uint[PGSZ / 4]; // 4 byte words
yield return 2;
for (var lowi = (PrimeT)0; ; lowi += BFBTS) {
for (var bi = 0; ; ++bi) {
if (bi < 1) {
if (bi < 0) { bi = 0; yield return 2; }
PrimeT nxt = 3 + lowi + lowi + BFRNG;
if (lowi <= 0) { // cull very first page
for (int i = 0, p = 3, sqr = 9; sqr < nxt; i++, p += 2, sqr = p * p)
if ((cbuf[i >> 5] & (1 << (i & 31))) == 0)
for (int j = (sqr - 3) >> 1; j < BFBTS; j += p)
cbuf[j >> 5] |= 1u << j;
}
else { // cull for the rest of the pages
Array.Clear(cbuf, 0, cbuf.Length);
if (bpa.Count == 0) { // inite secondar base primes stream
bps = nmrtr(); bps.MoveNext(); bps.MoveNext();
bpa.Add((uint)bps.Current); bps.MoveNext();
} // add 3 to base primes array
// make sure bpa contains enough base primes...
for (PrimeT p = bpa[bpa.Count - 1], sqr = p * p; sqr < nxt; ) {
p = bps.Current; bps.MoveNext(); sqr = p * p; bpa.Add((uint)p);
}
for (int i = 0, lmt = bpa.Count - 1; i < lmt; i++) {
var p = (PrimeT)bpa[i]; var s = (p * p - 3) >> 1;
// adjust start index based on page lower limit...
if (s >= lowi) s -= lowi;
else {
var r = (lowi - s) % p;
s = (r != 0) ? p - r : 0;
}
for (var j = (uint)s; j < BFBTS; j += p)
cbuf[j >> 5] |= 1u << ((int)j);
}
}
}
while (bi < BFBTS && (cbuf[bi >> 5] & (1 << (bi & 31))) != 0) ++bi;
if (bi < BFBTS) yield return 3 + (((PrimeT)bi + lowi) << 1);
else break; // outer loop for next page segment...
}
}
}
public IEnumerator<PrimeT> GetEnumerator() { return nmrtr(); }
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}

The above code is about 25 times faster than the Dictionary version at computing the first about 50 million primes (up to a range of one billion), with the actual enumeration of the result sequence now taking longer than the time it takes to cull the composite number representation bits from the arrays, meaning that it is over 50 times faster at actually sieving the primes. The code owes its speed as compared to a naive "one huge memory array" algorithm to using an array size that is the size of the CPU L1 or L2 caches and using bit-packing to fit more number representations into this limited capacity; in this way RAM memory access times are reduced by a factor of from about four to about 10 (depending on CPU and RAM speed) as compared to those naive implementations, and the minor computational cost of the bit manipulations is compensated by a large factor in total execution time.

The time to enumerate the result primes sequence can be reduced somewhat (about a second) by removing the automatic iterator "yield return" statements and converting them into a "rull-your-own" IEnumerable<PrimeT> implementation, but for page segmentation of odds-only, this iteration of the results will still take longer than the time to actually cull the composite numbers from the page arrays.

In order to make further gains in speed, custom methods must be used to avoid using iterator sequences. If this is done, then further gains can be made by extreme wheel factorization (up to about another about four times gain in speed) and multi-processing (with another gain in speed proportional to the actual independent CPU cores used).

Note that all of these gains in speed are not due to C# other than it compiles to reasonably efficient machine code, but rather to proper use of the Sieve of Eratosthenes algorithm.

All of the above unbounded code can be tested by the following "main" method (replace the name "PrimesXXX" with the name of the class to be tested):

    static void Main(string[] args) {
Console.WriteLine(PrimesXXX().ElementAt(1000000 - 1)); // zero based indexing...
}

To produce the following output for all tested versions (although some are considerably faster than others):

Output:
15485863

[edit] Chapel

This solution uses nested iterators to create new wheels at run time:

// yield prime and remove all multiples of it from children sieves
iter sieve(prime):int {
 
yield prime;
 
var last = prime;
label candidates for candidate in sieve(prime+1) do {
for composite in last..candidate by prime do {
 
// candidate is a multiple of this prime
if composite == candidate then {
// remember size of last composite
last = composite;
// and try the next candidate
continue candidates;
}
}
 
// candidate cannot need to be removed by this sieve
// yield to parent sieve for checking
yield candidate;
}
}
The topmost sieve needs to be started with 2 (the smallest prime):
config const N = 30;
for p in sieve(2) {
write(" ", p);
if p > N then {
writeln();
break;
}
}

[edit] Clojure

Calculates primes up to and including n using a mutable boolean array but otherwise entirely functional code.

 
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(let [root (-> n Math/sqrt long),
cmpsts (boolean-array (inc n)),
cullp (fn [p]
(loop [i (* p p)]
(if (<= i n)
(do (aset cmpsts i true)
(recur (+ i p))))))]
(do (dorun (map #(cullp %) (filter #(not (aget cmpsts %))
(range 2 (inc root)))))
(filter #(not (aget cmpsts %)) (range 2 (inc n))))))
 

Alternative implementation using Clojure's side-effect oriented list comprehension.

 
(defn primes-to
"Returns a lazy sequence of prime numbers less than lim"
[lim]
(let [refs (boolean-array (+ lim 1) true)
root (int (Math/sqrt lim))]
(do (doseq [i (range 2 lim)
 :while (<= i root)
 :when (aget refs i)]
(doseq [j (range (* i i) lim i)]
(aset refs j false)))
(filter #(aget refs %) (range 2 lim)))))
 

Alternative very slow entirely functional implementation using lazy sequences

 
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(letfn [(nxtprm [cs] ; current candidates
(let [p (first cs)]
(if (> p (Math/sqrt n)) cs
(cons p (lazy-seq (nxtprm (-> (range (* p p) (inc n) p)
set (remove cs) rest)))))))]
(nxtprm (range 2 (inc n)))))
 

The reason that the above code is so slow is that it has has a high constant factor overhead due to using a (hash) set to remove the composites from the future composites stream, each prime composite stream removal requires a scan across all remaining composites (compared to using an array or vector where only the culled values are referenced, and due to the slowness of Clojure sequence operations as compared to iterator/sequence operations in other languages.

Version based on immutable Vector's

Here is an immutable boolean vector based non-lazy sequence version other than for the lazy sequence operations to output the result:

 
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[max-prime]
(let [sieve (fn [s n]
(if (<= (* n n) max-prime)
(recur (if (s n)
(reduce #(assoc %1 %2 false) s (range (* n n) (inc max-prime) n))
s)
(inc n))
s))]
(->> (-> (reduce conj (vector-of :boolean) (map #(= % %) (range (inc max-prime))))
(assoc 0 false)
(assoc 1 false)
(sieve 2))
(map-indexed #(vector %2 %1)) (filter first) (map second))))
 

The above code is still quite slow due to the cost of the immutable copy-on-modify operations.

Odds only bit packed mutable array based version

The following code implements an odds-only sieve using a mutable bit packed long array, only using a lazy sequence for the output of the resulting primes:

 
(set! *unchecked-math* true)
 
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(let [root (-> n Math/sqrt long),
rootndx (long (/ (- root 3) 2)),
ndx (long (/ (- n 3) 2)),
cmpsts (long-array (inc (/ ndx 64))),
isprm #(zero? (bit-and (aget cmpsts (bit-shift-right % 6))
(bit-shift-left 1 (bit-and % 63)))),
cullp (fn [i]
(let [p (long (+ i i 3))]
(loop [i (bit-shift-right (- (* p p) 3) 1)]
(if (<= i ndx)
(do (let [w (bit-shift-right i 6)]
(aset cmpsts w (bit-or (aget cmpsts w)
(bit-shift-left 1 (bit-and i 63)))))
(recur (+ i p))))))),
cull (fn [] (loop [i 0] (if (<= i rootndx)
(do (if (isprm i) (cullp i)) (recur (inc i))))))]
(letfn [(nxtprm [i] (if (<= i ndx)
(cons (+ i i 3) (lazy-seq (nxtprm (loop [i (inc i)]
(if (or (> i ndx) (isprm i)) i
(recur (inc i)))))))))]
(if (< n 2) nil
(cons 3 (if (< n 3) nil (do (cull) (lazy-seq (nxtprm 0)))))))))
 

The above code is about as fast as any "one large sieving array" type of program in any computer language with this level of wheel factorization other than the lazy sequence operations are quite slow: it takes about ten times as long to enumerate the results as it does to do the actual sieving work of culling the composites from the sieving buffer array. The slowness of sequence operations is due to nested function calls, but primarily due to the way Clojure implements closures by "boxing" all arguments (and perhaps return values) as objects in the heap space, which then need to be "un-boxed" as primitives as necessary for integer operations. Some of the facilities provided by lazy sequences are not needed for this algorithm, such as the automatic memoization which means that each element of the sequence is calculated only once; it is not necessary for the sequence values to be retraced for this algorithm.

If further levels of wheel factorization were used, the time to enumerate the resulting primes would be an even higher overhead as compared to the actual composite number culling time, would get even higher if page segmentation were used to limit the buffer size to the size of the CPU L1 cache for many times better memory access times, most important in the culling operations, and yet higher again if multi-processing were used to share to page segment processing across CPU cores.

The following code overcomes many of those limitations by using an internal (OPSeq) "deftype" which implements the ISeq interface as well as the Counted interface to provide immediate count returns (based on a pre-computed total), as well as the IReduce interface which can greatly speed come computations based on the primes sequence (eased greatly using facilities provided by Clojure 1.7.0 and up):

 
(defn primes-tox
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(let [root (-> n Math/sqrt long),
rootndx (long (/ (- root 3) 2)),
ndx (max (long (/ (- n 3) 2)) 0),
lmt (quot ndx 64),
cmpsts (long-array (inc lmt)),
cullp (fn [i]
(let [p (long (+ i i 3))]
(loop [i (bit-shift-right (- (* p p) 3) 1)]
(if (<= i ndx)
(do (let [w (bit-shift-right i 6)]
(aset cmpsts w (bit-or (aget cmpsts w)
(bit-shift-left 1 (bit-and i 63)))))
(recur (+ i p))))))),
cull (fn [] (do (aset cmpsts lmt (bit-or (aget cmpsts lmt)
(bit-shift-left -2 (bit-and ndx 63))))
(loop [i 0]
(when (<= i rootndx)
(when (zero? (bit-and (aget cmpsts (bit-shift-right i 6))
(bit-shift-left 1 (bit-and i 63))))
(cullp i))
(recur (inc i))))))
numprms (fn []
(let [w (dec (alength cmpsts))] ;; fast results count bit counter
(loop [i 0, cnt (bit-shift-left (alength cmpsts) 6)]
(if (> i w) cnt
(recur (inc i)
(- cnt (java.lang.Long/bitCount (aget cmpsts i))))))))]
(if (< n 2) nil
(cons 2 (if (< n 3) nil
(do (cull)
(deftype OPSeq [^long i ^longs cmpsa ^long cnt ^long tcnt] ;; for arrays maybe need to embed the array so that it doesn't get garbage collected???
clojure.lang.ISeq
(first [_] (if (nil? cmpsa) nil (+ i i 3)))
(next [_] (let [ncnt (inc cnt)] (if (>= ncnt tcnt) nil
(OPSeq.
(loop [j (inc i)]
(let [p? (zero? (bit-and (aget cmpsa (bit-shift-right j 6))
(bit-shift-left 1 (bit-and j 63))))]
(if p? j (recur (inc j)))))
cmpsa ncnt tcnt))))
(more [this] (let [ncnt (inc cnt)] (if (>= ncnt tcnt) (OPSeq. 0 nil tcnt tcnt)
(.next this))))
(cons [this o] (clojure.core/cons o this))
(empty [_] (if (= cnt tcnt) nil (OPSeq. 0 nil tcnt tcnt)))
(equiv [this o] (if (or (not= (type this) (type o))
(not= cnt (.cnt ^OPSeq o)) (not= tcnt (.tcnt ^OPSeq o))
(not= i (.i ^OPSeq o))) false true))
clojure.lang.Counted
(count [_] (- tcnt cnt))
clojure.lang.Seqable
(clojure.lang.Seqable/seq [this] (if (= cnt tcnt) nil this))
clojure.lang.IReduce
(reduce [_ f v] (let [c (- tcnt cnt)]
(if (<= c 0) nil
(loop [ci i, n c, rslt v]
(if (zero? (bit-and (aget cmpsa (bit-shift-right ci 6))
(bit-shift-left 1 (bit-and ci 63))))
(let [rrslt (f rslt (+ ci ci 3)),
rdcd (reduced? rrslt),
nrslt (if rdcd @rrslt rrslt)]
(if (or (<= n 1) rdcd) nrslt
(recur (inc ci) (dec n) nrslt)))
(recur (inc ci) n rslt))))))
(reduce [this f] (if (nil? i) (f) (if (= (.count this) 1) (+ i i 3)
(.reduce ^clojure.lang.IReduce (.next this) f (+ i i 3)))))
clojure.lang.Sequential
Object
(toString [this] (if (= cnt tcnt) "()"
(.toString (seq (map identity this))))))
(->OPSeq 0 cmpsts 0 (numprms))))))))
 

'(time (count (primes-tox 10000000)))' takes about 40 milliseconds (compiled) to produce 664579.

Due to the better efficiency of the custom CIS type, the primes to the above range can be enumerated in about the same 40 milliseconds that it takes to cull and count the sieve buffer array.

Under Clojure 1.7.0, one can use '(time (reduce (fn [] (+ (long sum) (long v))) 0 (primes-tox 2000000)))' to find "142913828922" as the sum of the primes to two million as per Euler Problem 10 in about 40 milliseconds total with about half the time used for sieving the array and half for computing the sum.

To show how sensitive Clojure is to forms of expression of functions, the simple form '(time (reduce + (primes-tox 2000000)))' takes about twice as long even though it is using the same internal routine for most of the calculation due to the function not having the type coercion's.

Before one considers that this code is suitable for larger ranges, it is still lacks the improvements of page segmentation with pages about the size of the CPU L1/L2 caches (produces about a four times speed up), maximal wheel factorization (to make it another about four times faster), and the use of multi-processing (for a further gain of about 4 times for a multi-core desktop CPU such as an Intel i7), will make the sieving/counting code about 50 times faster than this, although there will only be a moderate improvement in the time to enumerate/process the resulting primes. Using these techniques, the number of primes to one billion can be counted in a small fraction of a second.

[edit] Unbounded Versions

For some types of problems such as finding the nth prime (rather than the sequence of primes up to m), a prime sieve with no upper bound is a better tool.

The following variations on an incremental Sieve of Eratosthenes are based on or derived from the Richard Bird sieve as described in the Epilogue of Melissa E. O'Neill's definitive paper:

A Clojure version of Richard Bird's Sieve using Lazy Sequences (sieves odds only)

 
(defn primes-Bird
"Computes the unbounded sequence of primes using a Sieve of Eratosthenes algorithm by Richard Bird."
[]
(letfn [(mltpls [p] (let [p2 (* 2 p)]
(letfn [(nxtmltpl [c]
(cons c (lazy-seq (nxtmltpl (+ c p2)))))]
(nxtmltpl (* p p))))),
(allmtpls [ps] (cons (mltpls (first ps)) (lazy-seq (allmtpls (next ps))))),
(union [xs ys] (let [xv (first xs), yv (first ys)]
(if (< xv yv) (cons xv (lazy-seq (union (next xs) ys)))
(if (< yv xv) (cons yv (lazy-seq (union xs (next ys))))
(cons xv (lazy-seq (union (next xs) (next ys)))))))),
(mrgmltpls [mltplss] (cons (first (first mltplss))
(lazy-seq (union (next (first mltplss))
(mrgmltpls (next mltplss)))))),
(minusStrtAt [n cmpsts] (loop [n n, cmpsts cmpsts]
(if (< n (first cmpsts))
(cons n (lazy-seq (minusStrtAt (+ n 2) cmpsts)))
(recur (+ n 2) (next cmpsts)))))]
(do (def oddprms (cons 3 (lazy-seq (let [cmpsts (-> oddprms (allmtpls) (mrgmltpls))]
(minusStrtAt 5 cmpsts)))))
(cons 2 (lazy-seq oddprms)))))
 

The above code is quite slow due to both that the data structure is a linear merging of prime multiples and due to the slowness of the Clojure sequence operations.

A Clojure version of the tree folding sieve using Lazy Sequences

The following code speeds up the above code by merging the linear sequence of sequences as above by pairs into a right-leaning tree structure:

 
(defn primes-treeFolding
"Computes the unbounded sequence of primes using a Sieve of Eratosthenes algorithm modified from Bird."
[]
(letfn [(mltpls [p] (let [p2 (* 2 p)]
(letfn [(nxtmltpl [c]
(cons c (lazy-seq (nxtmltpl (+ c p2)))))]
(nxtmltpl (* p p))))),
(allmtpls [ps] (cons (mltpls (first ps)) (lazy-seq (allmtpls (next ps))))),
(union [xs ys] (let [xv (first xs), yv (first ys)]
(if (< xv yv) (cons xv (lazy-seq (union (next xs) ys)))
(if (< yv xv) (cons yv (lazy-seq (union xs (next ys))))
(cons xv (lazy-seq (union (next xs) (next ys)))))))),
(pairs [mltplss] (let [tl (next mltplss)]
(cons (union (first mltplss) (first tl))
(lazy-seq (pairs (next tl)))))),
(mrgmltpls [mltplss] (cons (first (first mltplss))
(lazy-seq (union (next (first mltplss))
(mrgmltpls (pairs (next mltplss))))))),
(minusStrtAt [n cmpsts] (loop [n n, cmpsts cmpsts]
(if (< n (first cmpsts))
(cons n (lazy-seq (minusStrtAt (+ n 2) cmpsts)))
(recur (+ n 2) (next cmpsts)))))]
(do (def oddprms (cons 3 (lazy-seq (let [cmpsts (-> oddprms (allmtpls) (mrgmltpls))]
(minusStrtAt 5 cmpsts)))))
(cons 2 (lazy-seq oddprms)))))
 

The above code is still slower than it should be due to the slowness of Clojure's sequence operations.

A Clojure version of the above tree folding sieve using a custom Co Inductive Sequence

The following code uses a custom "deftype" non-memoizing Co Inductive Stream/Sequence (CIS) implementing the ISeq interface to make the sequence operations more efficient and is about four times faster than the above code:

 
(defn primes-treeFoldingx
"Computes the unbounded sequence of primes using a Sieve of Eratosthenes algorithm modified from Bird."
[]
(do (deftype CIS [v cont]
clojure.lang.ISeq
(first [_] v)
(next [_] (if (nil? cont) nil (cont)))
(more [this] (let [nv (.next this)] (if (nil? nv) (CIS. nil nil) nv)))
(cons [this o] (clojure.core/cons o this))
(empty [_] (if (and (nil? v) (nil? cont)) nil (CIS. nil nil)))
(equiv [this o] (loop [cis1 this, cis2 o] (if (nil? cis1) (if (nil? cis2) true false)
(if (or (not= (type cis1) (type cis2))
(not= (.v cis1) (.v ^CIS cis2))
(and (nil? (.cont cis1))
(not (nil? (.cont ^CIS cis2))))
(and (nil? (.cont ^CIS cis2))
(not (nil? (.cont cis1))))) false
(if (nil? (.cont cis1)) true
(recur ((.cont cis1)) ((.cont ^CIS cis2))))))))
(count [this] (loop [cis this, cnt 0] (if (or (nil? cis) (nil? (.cont cis))) cnt
(recur ((.cont cis)) (inc cnt)))))
clojure.lang.Seqable
(seq [this] (if (and (nil? v) (nil? cont)) nil this))
clojure.lang.Sequential
Object
(toString [this] (if (and (nil? v) (nil? cont)) "()" (.toString (seq (map identity this))))))
(letfn [(mltpls [p] (let [p2 (* 2 p)]
(letfn [(nxtmltpl [c]
(->CIS c (fn [] (nxtmltpl (+ c p2)))))]
(nxtmltpl (* p p))))),
(allmtpls [^CIS ps] (->CIS (mltpls (.v ps)) (fn [] (allmtpls ((.cont ps)))))),
(union [^CIS xs ^CIS ys] (let [xv (.v xs), yv (.v ys)]
(if (< xv yv) (->CIS xv (fn [] (union ((.cont xs)) ys)))
(if (< yv xv) (->CIS yv (fn [] (union xs ((.cont ys)))))
(->CIS xv (fn [] (union (next xs) ((.cont ys))))))))),
(pairs [^CIS mltplss] (let [^CIS tl ((.cont mltplss))]
(->CIS (union (.v mltplss) (.v tl))
(fn [] (pairs ((.cont tl))))))),
(mrgmltpls [^CIS mltplss] (->CIS (.v ^CIS (.v mltplss))
(fn [] (union ((.cont ^CIS (.v mltplss)))
(mrgmltpls (pairs ((.cont mltplss)))))))),
(minusStrtAt [n ^CIS cmpsts] (loop [n n, cmpsts cmpsts]
(if (< n (.v cmpsts))
(->CIS n (fn [] (minusStrtAt (+ n 2) cmpsts)))
(recur (+ n 2) ((.cont cmpsts))))))]
(do (def oddprms (->CIS 3 (fn [] (let [cmpsts (-> oddprms (allmtpls) (mrgmltpls))]
(minusStrtAt 5 cmpsts)))))
(->CIS 2 (fn [] oddprms))))))
 

'(time (count (take-while #(<= (long %) 10000000) (primes-treeFoldingx))))' takes about 3.4 seconds for a range of 10 million.

The above code is useful for ranges up to about fifteen million primes, which is about the first million primes; it is comparable in speed to all of the bounded versions except for the fastest bit packed version which can reasonably be used for ranges about 100 times as large.

Incremental Hash Map based unbounded "odds-only" version

The following code is a version of the O'Neill Haskell code but does not use wheel factorization other than for sieving odds only (although it could be easily added) and uses a Hash Map (constant amortized access time) rather than a Priority Queue (log n access time for combined remove-and-insert-anew operations, which are the majority used for this algorithm) with a lazy sequence for output of the resulting primes; the code has the added feature that it uses a secondary base primes sequence generator and only adds prime culling sequences to the composites map when they are necessary, thus saving time and limiting storage to only that required for the map entries for primes up to the square root of the currently sieved number:

 
(defn primes-hashmap
"Infinite sequence of primes using an incremental Sieve or Eratosthenes with a Hashmap"
[]
(letfn [(nxtoddprm [c q bsprms cmpsts]
(if (>= c q) ;; only ever equal
(let [p2 (* (first bsprms) 2), nbps (next bsprms), nbp (first nbps)]
(recur (+ c 2) (* nbp nbp) nbps (assoc cmpsts (+ q p2) p2)))
(if (contains? cmpsts c)
(recur (+ c 2) q bsprms
(let [adv (cmpsts c), ncmps (dissoc cmpsts c)]
(assoc ncmps
(loop [try (+ c adv)] ;; ensure map entry is unique
(if (contains? ncmps try)
(recur (+ try adv)) try)) adv)))
(cons c (lazy-seq (nxtoddprm (+ c 2) q bsprms cmpsts))))))]
(do (def baseoddprms (cons 3 (lazy-seq (nxtoddprm 5 9 baseoddprms {}))))
(cons 2 (lazy-seq (nxtoddprm 3 9 baseoddprms {}))))))
 

The above code is slower than the best tree folding version due to the added constant factor overhead of computing the hash functions for every hash map operation even though it has computational complexity of (n log log n) rather than the worse (n log n log log n) for the previous incremental tree folding sieve. It is still about 100 times slower than the sieve based on the bit-packed mutable array due to these constant factor hashing overheads.

There is almost no benefit of converting the above code to use a CIS as most of the time is expended in the hash map functions.

Incremental Priority Queue based unbounded "odds-only" version

In order to implement the O'Neill Priority Queue incremental Sieve of Eratosthenes algorithm, one requires an efficient implementation of a Priority Queue, which is not part of standard Clojure. For this purpose, the most suitable Priority Queue is a binary tree heap based MinHeap algorithm. The following code implements a purely functional (using entirely immutable state) MinHeap Priority Queue providing the required functions of (emtpy-pq) initialization, (getMin-pq pq) to examinte the minimum key/value pair in the queue, (insert-pq pq k v) to add entries to the queue, and (replaceMinAs-pq pq k v) to replaace the minimum entry with a key/value pair as given (it is more efficient that if functions were provided to delete and then re-insert entries in the queue; there is therefore no "delete" or other queue functions supplied as the algorithm does not requrie them:

 
(deftype PQEntry [k, v]
Object
(toString [_] (str "<" k "," v ">")))
(deftype PQNode [^PQEntry ntry, lft, rght, lvl]
Object
(toString [_] (str "<" lvl ntry " left: " (str lft) " right: " (str rght) ">")))
 
(defn empty-pq [] nil)
 
(defn getMin-pq ^PQEntry [pq] (condp instance? pq
PQEntry pq,
PQNode (.ntry ^PQNode pq)
nil))
 
(defn insert-pq [opq k v]
(loop [kv (->PQEntry k v), msk 0, pq opq, cont identity]
(condp instance? pq
PQEntry (if (< k (.k ^PQEntry pq)) (cont (->PQNode kv pq nil 2))
(cont (->PQNode pq kv nil 2))),
PQNode (let [^PQNode pqn pq, kvn (.ntry pqn), l (.lft pqn), r (.rght pqn),
nlvl (+ (.lvl pqn) 1),
nmsk (if (zero? msk) ;; never ever 0 again with the bit or'ed 1
(bit-or (bit-shift-left nlvl (- 64 (long (quot (Math/log (double nlvl))
(Math/log (double 2)))))) 1)
(bit-shift-left msk 1))]
(if (<= k (.k ^PQEntry kvn))
(if (neg? nmsk)
(recur kvn nmsk r (fn [npq] (cont (->PQNode kv l npq nlvl))))
(recur kvn nmsk l (fn [npq] (cont (->PQNode kv npq r nlvl)))))
(if (neg? nmsk)
(recur kv nmsk r (fn [npq] (cont (->PQNode kvn l npq nlvl))))
(recur kv nmsk l (fn [npq] (cont (->PQNode kvn npq r nlvl))))))),
(cont kv))))
 
(defn replaceMinAs-pq [opq k v]
(let [kv (->PQEntry k v)]
(loop [pq opq, cont identity]
(if (instance? PQNode pq)
(let [^PQNode pqn pq, l (.lft pqn), r (.rght pqn), lvl (.lvl pqn)]
(cond
(and (instance? PQEntry r) (> k (.k ^PQEntry r)))
(cond ;; right not empty so left is never empty
(and (instance? PQEntry l) (> k (.k ^PQEntry l))) ;; both qualify; choose least
(if (> (.k ^PQEntry l) (.k ^PQEntry r))
(cont (->PQNode r l kv lvl))
(cont (->PQNode l kv r lvl))),
(and (instance? PQNode l) (> k (.k ^PQEntry (.ntry ^PQNode l))))
(let [^PQEntry kvl (.ntry ^PQNode l)]
(if (> (.k kvl) (.k ^PQEntry r)) ;; both qualify; choose least
(cont (->PQNode r l kv lvl))
(recur l (fn [npq] (cont (->PQNode kvl npq r lvl)))))),
 :else (cont (->PQNode r l kv lvl))), ;; only right qualifies; no recursion
(and (instance? PQNode r) (> k (.k ^PQEntry (.ntry ^PQNode r))))
(let [^PQEntry kvr (.ntry ^PQNode r)]
(if (and (instance? PQNode l) (> k (.k ^PQEntry (.ntry ^PQNode l))))
(let [^PQEntry kvl (.ntry ^PQNode l)]
(if (> (.k kvl) (.k kvr)) ;; both qualify; choose least
(recur r (fn [npq] (cont (->PQNode kvr l npq lvl))))
(recur l (fn [npq] (cont (->PQNode kvl npq r lvl))))))
(recur r (fn [npq] (cont (->PQNode kvr l npq lvl)))))), ;; only right qualifies
 :else (cond ;; right is empty, but as this is a node, left is never empty
(and (instance? PQEntry l) (> k (.k ^PQEntry l)))
(cont (->PQNode l kv r lvl)),
(and (instance? PQNode l) (> k (.k ^PQEntry (.ntry ^PQNode l))))
(recur l (fn [npq] (cont (->PQNode (.ntry ^PQNode l) npq r lvl)))),
 :else (cont (->PQNode kv l r lvl))))) ;; just replace contents, leave same
(cont kv))))) ;; if was empty or just an entry, just use current entry
 

Note that the above code is written partially using continuation passing style so as to leave the "recur" calls in tail call position as required for efficient looping in Clojure; for practical sieving ranges, the algorithm could likely use just raw function recursion as recursion depth is unlikely to be used beyond a depth of about ten or so, but raw recursion is said to be less code efficient.

The actual incremental sieve using the Priority Queue is as follows, which code uses the same optimizations of postponing the addition of prime composite streams to the queue until the square root of the currently sieved number is reached and using a secondary base primes stream to generate the primes composite stream markers in the queue as was used for the Hash Map version:

 
(defn primes-pq
"Infinite sequence of primes using an incremental Sieve or Eratosthenes with a Priority Queue"
[]
(letfn [(nxtoddprm [c q bsprms cmpsts]
(if (>= c q) ;; only ever equal
(let [p2 (* (first bsprms) 2), nbps (next bsprms), nbp (first nbps)]
(recur (+ c 2) (* nbp nbp) nbps (insert-pq cmpsts (+ q p2) p2)))
(let [mn (getMin-pq cmpsts)]
(if (and mn (>= c (.k mn))) ;; never greater than
(recur (+ c 2) q bsprms
(loop [adv (.v mn), cmps cmpsts] ;; advance repeat composites for value
(let [ncmps (replaceMinAs-pq cmps (+ c adv) adv),
nmn (getMin-pq ncmps)]
(if (and nmn (>= c (.k nmn)))
(recur (.v nmn) ncmps)
ncmps))))
(cons c (lazy-seq (nxtoddprm (+ c 2) q bsprms cmpsts)))))))]
(do (def baseoddprms (cons 3 (lazy-seq (nxtoddprm 5 9 baseoddprms (empty-pq)))))
(cons 2 (lazy-seq (nxtoddprm 3 9 baseoddprms (empty-pq)))))))
 

The above code is faster than the Hash Map version up to about a sieving range of fifteen million or so, but gets progressively slower for larger ranges due to having (n log n log log n) computational complexity rather than the (n log log n) for the Hash Map version, which has a higher constant factor overhead that is overtaken by the extra "log n" factor.

It is slower that the fastest of the tree folding versions (which has the same computational complexity) due to the higher constant factor overhead of the Priority Queue operations (although perhaps a more efficient implementation of the MinHeap Priority Queue could be developed).

Again, these non-mutable array based sieves are about a hundred times slower than even the "one large memory buffer array" version as implemented in the bounded section; a page segmented version of the mutable bit-packed memory array would be several times faster.

All of these algorithms will respond to maximum wheel factorization, getting up to approximately four times faster if this is applied as compared to the the "odds-only" versions.

It is difficult if not impossible to apply efficient multi-processing to the above versions of the unbounded sieves as the next values of the primes sequence are dependent on previous changes of state for the Bird and Tree Folding versions; however, with the addition of a "update the whole Priority Queue (and reheapify)" or "update the Hash Map" to a given page start state functions, it is possible to do for these letter two algorithms; however, even though it is possible and there is some benefit for these latter two implementations, the benefit is less than using mutable arrays due to that the results must be enumerated into a data structure of some sort in order to be passed out of the page function whereas they can be directly enumerated from the array for the mutable array versions.

[edit] CMake

function(eratosthenes var limit)
# Check for integer overflow. With CMake using 32-bit signed integer,
# this check fails when limit > 46340.
if(NOT limit EQUAL 0) # Avoid division by zero.
math(EXPR i "(${limit} * ${limit}) / ${limit}")
if(NOT limit EQUAL ${i})
message(FATAL_ERROR "limit is too large, would cause integer overflow")
endif()
endif()
 
# Use local variables prime_2, prime_3, ..., prime_${limit} as array.
# Initialize array to y => yes it is prime.
foreach(i RANGE 2 ${limit})
set(prime_${i} y)
endforeach(i)
 
# Gather a list of prime numbers.
set(list)
foreach(i RANGE 2 ${limit})
if(prime_${i})
# Append this prime to list.
list(APPEND list ${i})
 
# For each multiple of i, set n => no it is not prime.
# Optimization: start at i squared.
math(EXPR square "${i} * ${i}")
if(NOT square GREATER ${limit}) # Avoid fatal error.
foreach(m RANGE ${square} ${limit} ${i})
set(prime_${m} n)
endforeach(m)
endif()
endif(prime_${i})
endforeach(i)
set(${var} ${list} PARENT_SCOPE)
endfunction(eratosthenes)
# Print all prime numbers through 100.
eratosthenes(primes 100)
message(STATUS "${primes}")

[edit] COBOL

*> Please ignore the asterisks in the first column of the next comments,
*> which are kludges to get syntax highlighting to work.
IDENTIFICATION DIVISION.
PROGRAM-ID. Sieve-Of-Eratosthenes.
 
DATA DIVISION.
WORKING-STORAGE SECTION.
 
01 Max-Number USAGE UNSIGNED-INT.
01 Max-Prime USAGE UNSIGNED-INT.
 
01 Num-Group.
03 Num-Table PIC X VALUE "P"
OCCURS 1 TO 10000000 TIMES DEPENDING ON Max-Number
INDEXED BY Num-Index.
88 Is-Prime VALUE "P" FALSE "N".
 
01 Current-Prime USAGE UNSIGNED-INT.
 
01 I USAGE UNSIGNED-INT.
 
PROCEDURE DIVISION.
DISPLAY "Enter the limit: " WITH NO ADVANCING
ACCEPT Max-Number
DIVIDE Max-Number BY 2 GIVING Max-Prime
 
* *> Set Is-Prime of all non-prime numbers to false.
SET Is-Prime (1) TO FALSE
PERFORM UNTIL Max-Prime < Current-Prime
* *> Set current-prime to next prime.
ADD 1 TO Current-Prime
PERFORM VARYING Num-Index FROM Current-Prime BY 1
UNTIL Is-Prime (Num-Index)
END-PERFORM
MOVE Num-Index TO Current-Prime
 
* *> Set Is-Prime of all multiples of current-prime to
* *> false, starting from current-prime sqaured.
COMPUTE Num-Index = Current-Prime ** 2
PERFORM UNTIL Max-Number < Num-Index
SET Is-Prime (Num-Index) TO FALSE
SET Num-Index UP BY Current-Prime
END-PERFORM
END-PERFORM
 
* *> Display the prime numbers.
PERFORM VARYING Num-Index FROM 1 BY 1
UNTIL Max-Number < Num-Index
IF Is-Prime (Num-Index)
DISPLAY Num-Index
END-IF
END-PERFORM
 
GOBACK
.

[edit] Common Lisp

(defun sieve-of-eratosthenes (maximum)
(let ((sieve (make-array (1+ maximum) :element-type 'bit
:initial-element 0)))
(loop for candidate from 2 to maximum
when (zerop (bit sieve candidate))
collect candidate
and do (loop for composite from (expt candidate 2)
to maximum by candidate
do (setf (bit sieve composite) 1)))))

Working with odds only (above twice speedup), and only test divide up to the square root of the maximum:

(defun sieve-odds (maximum) "sieve for odd numbers"
(cons 2
(let ((maxi (ash (1- maximum) -1)) (stop (ash (isqrt maximum) -1)))
(let ((sieve (make-array (1+ maxi) :element-type 'bit :initial-element 0)))
(loop for i from 1 to maxi
when (zerop (sbit sieve i))
collect (1+ (ash i 1))
and when (<= i stop) do
(loop for j from (ash (* i (1+ i)) 1) to maxi by (1+ (ash i 1))
do (setf (sbit sieve j) 1)))))))

While formally a wheel, odds are uniformly spaced and do not require any special processing except for value translation. Wheels proper aren't uniformly spaced and are thus trickier.

[edit] D

[edit] Simpler Version

Prints all numbers less than the limit.
import std.stdio, std.algorithm, std.range, std.functional;
 
uint[] sieve(in uint limit) nothrow @safe {
if (limit < 2)
return [];
auto composite = new bool[limit];
 
foreach (immutable uint n; 2 .. cast(uint)(limit ^^ 0.5) + 1)
if (!composite[n])
for (uint k = n * n; k < limit; k += n)
composite[k] = true;
 
//return iota(2, limit).filter!(not!composite).array;
return iota(2, limit).filter!(i => !composite[i]).array;
}
 
void main() {
50.sieve.writeln;
}
Output:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]

[edit] Faster Version

This version uses an array of bits (instead of booleans, that are represented with one byte), and skips even numbers. The output is the same.

import std.stdio, std.math, std.array;
 
size_t[] sieve(in size_t m) pure nothrow @safe {
if (m < 3)
return null;
immutable size_t n = m - 1;
enum size_t bpc = size_t.sizeof * 8;
auto F = new size_t[((n + 2) / 2) / bpc + 1];
F[] = size_t.max;
 
size_t isSet(in size_t i) nothrow @safe @nogc {
immutable size_t offset = i / bpc;
immutable size_t mask = 1 << (i % bpc);
return F[offset] & mask;
}
 
void resetBit(in size_t i) nothrow @safe @nogc {
immutable size_t offset = i / bpc;
immutable size_t mask = 1 << (i % bpc);
if ((F[offset] & mask) != 0)
F[offset] = F[offset] ^ mask;
}
 
for (size_t i = 3; i <= sqrt(real(n)); i += 2)
if (isSet((i - 3) / 2))
for (size_t j = i * i; j <= n; j += 2 * i)
resetBit((j - 3) / 2);
 
Appender!(size_t[]) result;
result ~= 2;
for (size_t i = 3; i <= n; i += 2)
if (isSet((i - 3) / 2))
result ~= i;
return result.data;
}
 
void main() {
50.sieve.writeln;
}

[edit] Extensible Version

(This version is used in the task Extensible prime generator.)

/// Extensible Sieve of Eratosthenes.
struct Prime {
uint[] a = [2];
 
private void grow() pure nothrow @safe {
immutable p0 = a[$ - 1] + 1;
auto b = new bool[p0];
 
foreach (immutable di; a) {
immutable uint i0 = p0 / di * di;
uint i = (i0 < p0) ? i0 + di - p0 : i0 - p0;
for (; i < b.length; i += di)
b[i] = true;
}
 
foreach (immutable uint i, immutable bi; b)
if (!b[i])
a ~= p0 + i;
}
 
uint opCall(in uint n) pure nothrow @safe {
while (n >= a.length)
grow;
return a[n];
}
}
 
version (sieve_of_eratosthenes3_main) {
void main() {
import std.stdio, std.range, std.algorithm;
 
Prime prime;
uint.max.iota.map!prime.until!q{a > 50}.writeln;
}
}

To see the output (that is the same), compile with -version=sieve_of_eratosthenes3_main.

[edit] Dart

// helper function to pretty print an Iterable
String iterableToString(Iterable seq) {
String str = "[";
Iterator i = seq.iterator;
if (i.moveNext()) str += i.current.toString();
while(i.moveNext()) {
str += ", " + i.current.toString();
}
return str + "]";
}
 
main() {
int limit = 1000;
int strt = new DateTime.now().millisecondsSinceEpoch;
Set<int> sieve = new Set<int>();
 
for(int i = 2; i <= limit; i++) {
sieve.add(i);
}
for(int i = 2; i * i <= limit; i++) {
if(sieve.contains(i)) {
for(int j = i * i; j <= limit; j += i) {
sieve.remove(j);
}
}
}
var sortedValues = new List<int>.from(sieve);
int elpsd = new DateTime.now().millisecondsSinceEpoch - strt;
print("Found " + sieve.length.toString() + " primes up to " + limit.toString() +
" in " + elpsd.toString() + " milliseconds.");
print(iterableToString(sortedValues)); // expect sieve.length to be 168 up to 1000...
// Expect.equals(168, sieve.length);
}
Output:

Found 168 primes up to 1000 in 9 milliseconds. [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]

Although it has the characteristics of a true Sieve of Eratosthenes, the above code isn't very efficient due to the remove/modify operations on the Set. Due to these, the computational complexity isn't close to linear with increasing range and it is quite slow for larger sieve ranges compared to compiled languages, taking about four seconds to sieve to ten million.

[edit] faster bit-packed array odds-only solution

import 'dart:math';
 
List<int> SoEOdds(int limit) {
List<int> prms = new List();
if (limit < 2) return prms;
prms.add(2);
if (limit < 3) return prms;
int lmt = (limit - 3) >> 1;
int bfsz = (lmt >> 5) + 1;
int sqrtlmt = (sqrt(limit) - 3).floor() >> 1;
var buf = new List<int>();
for (int i = 0; i < bfsz; i++)
buf.add(0);
for (int i = 0; i <= sqrtlmt; i++)
if ((buf[i >> 5] & (1 << (i & 31))) == 0) {
int p = i + i + 3;
for (int j = (p * p - 3) >> 1; j <= lmt; j += p)
buf[j >> 5] |= 1 << (j & 31);
}
for (int i = 0; i <= lmt; i++)
if ((buf[i >> 5] & (1 << (i & 31))) == 0)
prms.add(i + i + 3);
return prms;
}
 
void main() {
int limit = 10000000;
int strt = new DateTime.now().millisecondsSinceEpoch;
List<int> primes = SoEOdds(limit);
int count = primes.length;
int elpsd = new DateTime.now().millisecondsSinceEpoch - strt;
print("Found " + count.toString() + " primes up to " + limit.toString() +
" in " + elpsd.toString() + " milliseconds.");
// print(iterableToString(primes)); // expect sieve.length to be 168 up to 1000...
}

The above code is somewhat faster at about ten seconds using the Dart VM to sieve to 100 million, although much faster at about 1.5 seconds run conventionally in Google Chrome using the JavaScript V8 engine, likely due to JavaScript using double floating point numbers for int's whereas the Dart VM uses arbitrary precision integers.

[edit] fast page segmented array infinite iterator (sieves odds-only)

Translation of: JavaScript
import 'dart:collection';
 
class _SoEPagedIterator implements Iterator<int> {
static const int _BFSZ = 1 << 16;
static const int _BFBTS = _BFSZ * 32;
static const int _BFRNG = _BFBTS * 2;
int _prime = null;
int _bi = -1;
int _lowi = 0;
List<int> _bpa = new List<int>();
Iterator<int> _bps;
List<int> _buf = new List<int>();
int get current => this._prime;
bool moveNext() {
// the following redundant local variable declaration is necessary to
// prevent the dart2js compiler from "tree-shaking" and eliminating some
// essential code from the below, which doesn't happen with the Dart VM compiler.
int lowi = this._lowi;
while (true) {
if (this._bi < 1) {
if (this._bi < 0) { this._bi++; this._prime = 2; break; }
int nxt = 3 + (this._lowi << 1) + _BFRNG;
this._buf.clear();
for (int i = 0; i < _BFSZ; i++) this._buf.add(0); // faster initialization:
if (lowi <= 0) { // special culling for first page as no base primes yet:
for (int i = 0, p = 3, sqr = 9; sqr < nxt; i++, p += 2, sqr = p * p)
if ((this._buf[i >> 5] & (1 << (i & 31))) == 0)
for (int j = (sqr - 3) >> 1; j < _BFBTS; j += p)
this._buf[j >> 5] |= 1 << (j & 31);
} else { // after the first page:
if (this._bpa.length == 0) { // if this is the first page after the zero one:
this._bps = new _SoEPagedIterator(); // initialize separate base primes stream:
this._bps.moveNext(); // advance to the only even prime of two
this._bps.moveNext(); // advance past 2 to the next prime of 3
}
// get enough base primes for the page range...
for (var lp = this._bps.current, sqr = lp * lp; sqr < nxt;
this._bps.moveNext(), lp = this._bps.current, sqr = lp * lp) this._bpa.add(lp);
for (var i = 0; i < this._bpa.length; i++) {
int p = this._bpa[i];
int s = (p * p - 3) >> 1;
if (s >= this._lowi) // adjust start index based on page lower limit...
s -= this._lowi;
else {
int r = (this._lowi - s) % p;
s = (r != 0) ? p - r : 0;
}
for (var j = s; j < _BFBTS; j += p)
this._buf[j >> 5] |= 1 << (j & 31);
}
}
}
while (this._bi < _BFBTS && ((this._buf[this._bi >> 5] & (1 << (this._bi & 31))) != 0))
this._bi++; // find next marker still with prime status
if (this._bi < _BFBTS) { // within buffer: output computed prime
this._prime = 3 + ((this._lowi + this._bi++) << 1); break; }
else { // beyond buffer range: advance buffer
this._bi = 0;
this._lowi += _BFBTS;
lowi = this._lowi;
}
} return true;
}
}
 
class SoEPagedOddsInfGen extends IterableBase<int> {
Iterator<int> get iterator { return new _SoEPagedIterator(); }
}
 
void main() {
int n = 1000000000;
int strt = new DateTime.now().millisecondsSinceEpoch;
int count = new SoEPagedOddsInfGen().takeWhile((p) => p <= n).length;
int elpsd = new DateTime.now().millisecondsSinceEpoch - strt;
print("For a range of " + n.toString() + ", " + count.toString() +
" primes found in " + elpsd.toString() + " milliseconds.");
}

This version calculates the 50,847,534 primes up to one billion in about 20 seconds under the Dart Virtual Machine (VM). Under the Google Chrome V8 JavaScript engine it should take the same time as the JavaScript from which it was translated of about five seconds, but takes about 14 seconds due to the dart2js compiler adding extra run time array buffer range checks to the innermost culling loops, even though the "check" compiler option was not selected.

Also note the comment at the beginning of the "moveNext()" method about the redundant local variable needed to be added in order for the code to run under JavaScript using Dart 1.5.1 (and possible other versions), which shouldn't happen when it runs fine under the Dart VM without that extra local variable (based only on the private class field _lowi).

[edit] Delphi

program erathostenes;
 
{$APPTYPE CONSOLE}
 
type
TSieve = class
private
fPrimes: TArray<boolean>;
procedure InitArray;
procedure Sieve;
function getNextPrime(aStart: integer): integer;
function getPrimeArray: TArray<integer>;
public
function getPrimes(aMax: integer): TArray<integer>;
end;
 
{ TSieve }
 
function TSieve.getNextPrime(aStart: integer): integer;
begin
result := aStart;
while not fPrimes[result] do
inc(result);
end;
 
function TSieve.getPrimeArray: TArray<integer>;
var
i, n: integer;
begin
n := 0;
setlength(result, length(fPrimes)); // init array with maximum elements
for i := 2 to high(fPrimes) do
begin
if fPrimes[i] then
begin
result[n] := i;
inc(n);
end;
end;
setlength(result, n); // reduce array to actual elements
end;
 
function TSieve.getPrimes(aMax: integer): TArray<integer>;
begin
setlength(fPrimes, aMax);
InitArray;
Sieve;
result := getPrimeArray;
end;
 
procedure TSieve.InitArray;
begin
for i := 2 to high(fPrimes) do
fPrimes[i] := true;
end;
 
procedure TSieve.Sieve;
var
i, n, max: integer;
begin
max := length(fPrimes);
i := 2;
while i < sqrt(max) do
begin
n := sqr(i);
while n < max do
begin
fPrimes[n] := false;
inc(n, i);
end;
i := getNextPrime(i + 1);
end;
end;
 
var
i: integer;
Sieve: TSieve;
 
begin
Sieve := TSieve.Create;
for i in Sieve.getPrimes(100) do
write(i, ' ');
Sieve.Free;
readln;
end.

Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 


[edit] DWScript

function Primes(limit : Integer) : array of Integer;
var
n, k : Integer;
sieve := new Boolean[limit+1];
begin
for n := 2 to Round(Sqrt(limit)) do begin
if not sieve[n] then begin
for k := n*n to limit step n do
sieve[k] := True;
end;
end;
 
for k:=2 to limit do
if not sieve[k] then
Result.Add(k);
end;
 
var r := Primes(50);
var i : Integer;
for i:=0 to r.High do
PrintLn(r[i]);

[edit] Dylan

With outer to sqrt and inner to p^2 optimizations:

define method primes(n)
let limit = floor(n ^ 0.5) + 1;
let sieve = make(limited(<simple-vector>, of: <boolean>), size: n + 1, fill: #t);
let last-prime = 2;
 
while (last-prime < limit)
for (x from last-prime ^ 2 to n by last-prime)
sieve[x] := #f;
end for;
block (found-prime)
for (n from last-prime + 1 below limit)
if (sieve[n] = #f)
last-prime := n;
found-prime()
end;
end;
last-prime := limit;
end block;
end while;
 
for (x from 2 to n)
if (sieve[x]) format-out("Prime: %d\n", x); end;
end;
end;


[edit] E

E's standard library doesn't have a step-by-N numeric range, so we'll define one, implementing the standard iteration protocol.

def rangeFromBelowBy(start, limit, step) {
  return def stepper {
    to iterate(f) {
      var i := start
      while (i < limit) {
        f(null, i)
        i += step
      }
    }
  }
}

The sieve itself:

def eratosthenes(limit :(int > 2), output) {
  def composite := [].asSet().diverge()
  for i ? (!composite.contains(i)) in 2..!limit {
    output(i)
    composite.addAll(rangeFromBelowBy(i ** 2, limit, i))
  }
}

Example usage:

? eratosthenes(12, println)
# stdout: 2
#         3
#         5
#         7
#         11

[edit] eC

This example is incorrect. It uses rem testing and so is a trial division algorithm, not a sieve of Eratosthenes. Please fix the code and remove this message.

Note: this is not a Sieve of Eratosthenes; it is just trial division.

 
public class FindPrime
{
Array<int> primeList { [ 2 ], minAllocSize = 64 };
int index;
 
index = 3;
 
bool HasPrimeFactor(int x)
{
int max = (int)floor(sqrt((double)x));
 
for(i : primeList)
{
if(i > max) break;
if(x % i == 0) return true;
}
return false;
}
 
public int GetPrime(int x)
{
if(x > primeList.count - 1)
{
for (; primeList.count != x; index += 2)
if(!HasPrimeFactor(index))
{
if(primeList.count >= primeList.minAllocSize) primeList.minAllocSize *= 2;
primeList.Add(index);
}
}
return primeList[x-1];
}
}
 
class PrimeApp : Application
{
FindPrime fp { };
void Main()
{
int num = argc > 1 ? atoi(argv[1]) : 1;
PrintLn(fp.GetPrime(num));
}
}
 

[edit] Eiffel

Works with: EiffelStudio version 6.6 beta (with provisional loop syntax)
class
APPLICATION
 
create
make
 
feature
make
-- Run application.
do
across primes_through (100) as ic loop print (ic.item.out + " ") end
end
 
primes_through (a_limit: INTEGER): LINKED_LIST [INTEGER]
-- Prime numbers through `a_limit'
require
valid_upper_limit: a_limit >= 2
local
l_tab: ARRAY [BOOLEAN]
do
create Result.make
create l_tab.make_filled (True, 2, a_limit)
across
l_tab as ic
loop
if ic.item then
Result.extend (ic.target_index)
across ((ic.target_index * ic.target_index) |..| l_tab.upper).new_cursor.with_step (ic.target_index) as id
loop
l_tab [id.item] := False
end
end
end
end
end

Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

[edit] Elixir

This example is incorrect. This code uses rem testing and so is a trial division algorithm, not a sieve of Eratosthenes. Please fix the code and remove this message.
defmodule Prime do
def eratosthenes(limit \\ 1000) do
seq = for i <- 2..limit, do: i
sieve(seq, [])
end
 
defp sieve([], sieved), do: Enum.reverse(sieved)
defp sieve([h | t], sieved) do
list = for x <- t, rem(x,h)!=0, do: x
sieve(list, [h | sieved])
end
end
 
IO.inspect Prime.eratosthenes(100)
Output:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

[edit] Emacs Lisp

 
(defun sieve-set (limit)
(let ((xs (make-vector (1+ limit) 0)))
(loop for i from 2 to limit
when (zerop (aref xs i))
collect i
and do (loop for m from (* i i) to limit by i
do (aset xs m 1)))))
 

Straightforward implementation of sieve of Eratosthenes, 2 times faster:

 
(defun sieve (limit)
(let ((xs (vconcat [0 0] (number-sequence 2 limit))))
(loop for i from 2 to (sqrt limit)
when (aref xs i)
do (loop for m from (* i i) to limit by i
do (aset xs m 0)))
(remove 0 xs)))
 

[edit] Erlang

 
-module( sieve_of_eratosthenes ).
 
-export( [primes_upto/1] ).
 
primes_upto( N ) ->
Ns = lists:seq( 2, N ),
Dict = dict:from_list( [{X, potential_prime} || X <- Ns] ),
{Upto_sqrt_ns, _T} = lists:split( erlang:round(math:sqrt(N)), Ns ),
{N, Prime_dict} = lists:foldl( fun find_prime/2, {N, Dict}, Upto_sqrt_ns ),
lists:sort( dict:fetch_keys(Prime_dict) ).
 
 
 
find_prime( N, {Max, Dict} ) -> find_prime( dict:find(N, Dict), N, {Max, Dict} ).
 
find_prime( error, _N, Acc ) -> Acc;
find_prime( {ok, _Value}, N, {Max, Dict} ) -> {Max, lists:foldl( fun dict:erase/2, Dict, lists:seq(N*N, Max, N) )}.
 
Output:
35> sieve_of_eratosthenes:primes_upto( 20 ).
[2,3,5,7,11,13,17,19]

[edit] ERRE

 
PROGRAM SIEVE_ORG
 ! --------------------------------------------------
 ! Eratosthenes Sieve Prime Number Program in BASIC
 ! (da 3 a SIZE*2) from Byte September 1981
 !---------------------------------------------------
CONST SIZE%=8190
 
DIM FLAGS%[SIZE%]
 
BEGIN
PRINT("Only 1 iteration")
COUNT%=0
FOR I%=0 TO SIZE% DO
IF FLAGS%[I%]=TRUE THEN
 !$NULL
ELSE
PRIME%=I%+I%+3
K%=I%+PRIME%
WHILE NOT (K%>SIZE%) DO
FLAGS%[K%]=TRUE
K%=K%+PRIME%
END WHILE
PRINT(PRIME%;)
COUNT%=COUNT%+1
END IF
END FOR
PRINT
PRINT(COUNT%;" PRIMES")
END PROGRAM
 
Output:

last lines of the output screen

 15749  15761  15767  15773  15787  15791  15797  15803  15809  15817  15823 
 15859  15877  15881  15887  15889  15901  15907  15913  15919  15923  15937 
 15959  15971  15973  15991  16001  16007  16033  16057  16061  16063  16067 
 16069  16073  16087  16091  16097  16103  16111  16127  16139  16141  16183 
 16187  16189  16193  16217  16223  16229  16231  16249  16253  16267  16273 
 16301  16319  16333  16339  16349  16361  16363  16369  16381 
 1899  PRIMES

[edit] Euphoria

constant limit = 1000
sequence flags,primes
flags = repeat(1, limit)
for i = 2 to sqrt(limit) do
if flags[i] then
for k = i*i to limit by i do
flags[k] = 0
end for
end if
end for
 
primes = {}
for i = 2 to limit do
if flags[i] = 1 then
primes &= i
end if
end for
? primes

Output:

{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,
181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,
277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,
383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,
487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,
601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,
709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,
827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,
947,953,967,971,977,983,991,997}

[edit] F#

[edit] Functional

Richard Bird Sieve

This is the idea behind Richard Bird's unbounded code presented in the Epilogue of M. O'Neill's article in Haskell. It is about twice as much code as the Haskell code because F# does not have a built-in lazy list so that the effect must be constructed using a Co-Inductive Stream (CIS) type since no memoization is required, along with the use of recursive functions in combination with sequences. The type inference needs some help with the new CIS type (including selecting the generic type for speed). Note the use of recursive functions to implement multiple non-sharing delayed generating base primes streams, which along with these being non-memoizing means that the entire primes stream is not held in memory as for the original Bird code:

type CIS<'T> = struct val v:'T val cont:unit->CIS<'T> //'Co Inductive Stream for laziness
new (v,cont) = { v = v; cont = cont } end
type Primes = uint32
 
let primesBird() =
let rec (^^) (xs: CIS<Prime>) (ys: CIS<Prime>) = // stream merge function
let x = xs.v in let y = ys.v
if x < y then CIS(x, fun() -> xs.cont() ^^ ys)
elif y < x then CIS(y, fun() -> xs ^^ ys.cont())
else CIS(x, fun() -> xs.cont() ^^ ys.cont()) // no duplications
let pmltpls p = let rec nxt c = CIS(c, fun() -> nxt (c + p)) in nxt (p * p)
let rec allmltps (ps: CIS<Prime>) = CIS(pmltpls ps.v, fun() -> allmltps (ps.cont()))
let rec cmpsts (css: CIS<CIS<Prime>>) =
CIS(css.v.v, fun() -> (css.v.cont()) ^^ (cmpsts (css.cont())))
let rec minusat n (cs: CIS<Prime>) =
if n < cs.v then CIS(n, fun() -> minusat (n + 1u) cs)
else minusat (n + 1u) (cs.cont())
let rec baseprms() = CIS(2u, fun() -> minusat 3u (cmpsts (allmltps (baseprms()))))
Seq.unfold (fun (ps: CIS<Prime>) -> Some(ps.v, ps.cont()))
(minusat 2u (cmpsts (allmltps (baseprms()))))

The above code sieves all numbers of two and up including all even numbers as per the page specification; the following code makes the very minor changes for an odds-only sieve, with a speedup of over a factor of two:

type CIS<'T> = struct val v:'T val cont:unit->CIS<'T> //'Co Inductive Stream for laziness
new (v,cont) = { v = v; cont = cont } end
type Prime = uint32
 
let primesBirdOdds() =
let rec (^^) (xs: CIS<Prime>) (ys: CIS<Prime>) = // stream merge function
let x = xs.v in let y = ys.v
if x < y then CIS(x, fun() -> xs.cont() ^^ ys)
elif y < x then CIS(y, fun() -> xs ^^ ys.cont())
else CIS(x, fun() -> xs.cont() ^^ ys.cont()) // no duplications
let pmltpls p = let adv = p + p
let rec nxt c = CIS(c, fun() -> nxt (c + adv)) in nxt (p * p)
let rec allmltps (ps: CIS<Prime>) = CIS(pmltpls ps.v, fun() -> allmltps (ps.cont()))
let rec cmpsts (css: CIS<CIS<Prime>>) =
CIS(css.v.v, fun() -> (css.v.cont()) ^^ (cmpsts (css.cont())))
let rec minusat n (cs: CIS<Prime>) =
if n < cs.v then CIS(n, fun() -> minusat (n + 2u) cs)
else minusat (n + 2u) (cs.cont())
let rec oddprms() = CIS(3u, fun() -> minusat 5u (cmpsts (allmltps (oddprms()))))
Seq.unfold (fun (ps: CIS<Prime>) -> Some(ps.v, ps.cont()))
(CIS(2u, fun() -> minusat 3u (cmpsts (allmltps (oddprms())))))

Tree Folding Sieve

The above code is still somewhat inefficient as it operates on a linear right extending structure that deepens linearly with increasing base primes (those up to the square root of the currently sieved number); the following code changes the structure into an infinite binary tree-like folding by combining each pair of prime composite streams before further processing as usual - this decreases the processing by approximately a factor of log n:

type CIS<'T> = struct val v:'T val cont:unit->CIS<'T> //'Co Inductive Stream for laziness
new (v,cont) = { v = v; cont = cont } end
type Prime = uint32
 
let primesTreeFold() =
let rec (^^) (xs: CIS<Prime>) (ys: CIS<Prime>) = // merge streams; no duplicates
let x = xs.v in let y = ys.v
if x < y then CIS(x, fun() -> xs.cont() ^^ ys)
elif y < x then CIS(y, fun() -> xs ^^ ys.cont())
else CIS(x, fun() -> xs.cont() ^^ ys.cont())
let pmltpls p = let adv = p + p
let rec nxt c = CIS(c, fun() -> nxt (c + adv)) in nxt (p * p)
let rec allmltps (ps: CIS<Prime>) = CIS(pmltpls ps.v, fun() -> allmltps (ps.cont()))
let rec pairs (css: CIS<CIS<Prime>>) =
let ncss = css.cont()
CIS(css.v ^^ ncss.v, fun() -> pairs (ncss.cont()))
let rec cmpsts (css: CIS<CIS<Prime>>) =
CIS(css.v.v, fun() -> (css.v.cont()) ^^ (cmpsts << pairs << css.cont)())
let rec minusat n (cs: CIS<Prime>) =
if n < cs.v then CIS(n, fun() -> minusat (n + 2u) cs)
else minusat (n + 2u) (cs.cont())
let rec oddprms() = CIS(3u, fun() -> (minusat 5u << cmpsts << allmltps) (oddprms()))
Seq.unfold (fun (ps: CIS<Prime>) -> Some(ps.v, ps.cont()))
(CIS(2u, fun() -> (minusat 3u << cmpsts << allmltps) (oddprms())))

The above code is over four times faster than the "BirdOdds" version and is moderately useful for a range of the first million primes or so.

Priority Queue Sieve

In order to investigate Priority Queue Sieves as espoused by O'Neill in the referenced article, one must find an equivalent implementation of a Min Heap Priority Queue as used by her. There is such an purely functional implementation in RosettaCode translated from the Haskell code she used, from which the essential parts are duplicated here (Note that the key value is given an integer type in order to avoid the inefficiency of F# in generic comparison):

[<RequireQualifiedAccess>]
module MinHeap =
 
type HeapEntry<'V> = struct val k:uint32 val v:'V new(k,v) = {k=k;v=v} end
[<CompilationRepresentation(CompilationRepresentationFlags.UseNullAsTrueValue)>]
[<NoEquality; NoComparison>]
type PQ<'V> =
| Mt
| Br of HeapEntry<'
V> * PQ<'V> * PQ<'V>
 
let empty = Mt
 
let peekMin = function | Br(kv, _, _) -> Some(kv.k, kv.v)
| _ -> None
 
let rec push wk wv =
function | Mt -> Br(HeapEntry(wk, wv), Mt, Mt)
| Br(vkv, ll, rr) ->
if wk <= vkv.k then
Br(HeapEntry(wk, wv), push vkv.k vkv.v rr, ll)
else Br(vkv, push wk wv rr, ll)
 
let private siftdown wk wv pql pqr =
let rec sift pl pr =
match pl with
| Mt -> Br(HeapEntry(wk, wv), Mt, Mt)
| Br(vkvl, pll, plr) ->
match pr with
| Mt -> if wk <= vkvl.k then Br(HeapEntry(wk, wv), pl, Mt)
else Br(vkvl, Br(HeapEntry(wk, wv), Mt, Mt), Mt)
| Br(vkvr, prl, prr) ->
if wk <= vkvl.k && wk <= vkvr.k then Br(HeapEntry(wk, wv), pl, pr)
elif vkvl.k <= vkvr.k then Br(vkvl, sift pll plr, pr)
else Br(vkvr, pl, sift prl prr)
sift pql pqr
 
let replaceMin wk wv = function | Mt -> Mt
| Br(_, ll, rr) -> siftdown wk wv ll rr

Except as noted for any individual code, all of the following codes need the following prefix code in order to implement the non-memoizing Co-Inductive Streams (CIS's) and to set the type of particular constants used in the codes to the same time as the "Prime" type:

type CIS<'T> = struct val v: 'T val cont: unit -> CIS<'T> new(v,cont) = {v=v;cont=cont} end
type Prime = uint32
let frstprm = 2u
let frstoddprm = 3u
let inc1 = 1u
let inc = 2u

The F# equivalent to O'Neill's "odds-only" code is then implemented as follows, which needs the included changed prefix in order to change the primes type to a larger one to prevent overflow (as well the key type for the MinHeap needs to be changed from uint32 to uint64); it is functionally the same as the O'Neill code other than for minor changes to suit the use of CIS streams and the option output of the "peekMin" function:

type CIS<'T> = struct val v: 'T val cont: unit -> CIS<'T> new(v,cont) = {v=v;cont=cont} end
type Prime = uint64
let frstprm = 2UL
let frstoddprm = 3UL
let inc = 2UL
 
let primesPQ() =
let pmult p (xs: CIS<Prime>) = // does map (* p) xs
let rec nxtm (cs: CIS<Prime>) =
CIS(p * cs.v, fun() -> nxtm (cs.cont())) in nxtm xs
let insertprime p xs table =
MinHeap.push (p * p) (pmult p xs) table
let rec sieve'
(ns: CIS<Prime>) table =
let nextcomposite = match MinHeap.peekMin table with
| None -> ns.v // never happens
| Some (k, _) -> k
let rec adjust table =
let (n, advs) = match MinHeap.peekMin table with
| None -> (ns.v, ns.cont()) // never happens
| Some kv -> kv
if n <= ns.v then adjust (MinHeap.replaceMin advs.v (advs.cont()) table)
else table
if nextcomposite <= ns.v then sieve' (ns.cont()) (adjust table)
else let n = ns.v in CIS(n, fun() ->
let nxtns = ns.cont() in sieve'
nxtns (insertprime n nxtns table))
let rec sieve (ns: CIS<Prime>) = let n = ns.v in CIS(n, fun() ->
let nxtns = ns.cont() in sieve' nxtns (insertprime n nxtns MinHeap.empty))
let odds = // is the odds CIS from 3 up
let rec nxto i = CIS(i, fun() -> nxto (i + inc)) in nxto frstoddprm
Seq.unfold (fun (cis: CIS<Prime>) -> Some(cis.v, cis.cont()))
(CIS(frstprm, fun() -> (sieve odds)))

However, that algorithm suffers in speed and memory use due to over-eager adding of prime composite streams to the queue such that the queue used is much larger than it needs to be and a much larger range of primes number must be used in order to avoid numeric overflow on the square of the prime added to the queue. The following code corrects that by using a secondary (actually a multiple of) base primes streams which are constrained to be based on a prime that is no larger than the square root of the currently sieved number - this permits the use of much smaller Prime types as per the default prefix:

let primesPQx() =
let rec nxtprm n pq q (bps: CIS<Prime>) =
if n >= q then let bp = bps.v in let adv = bp + bp
let nbps = bps.cont() in let nbp = nbps.v
nxtprm (n + inc) (MinHeap.push (n + adv) adv pq) (nbp * nbp) nbps
else let ck, cv = match MinHeap.peekMin pq with
| None -> (q, inc) // only happens until first insertion
| Some kv -> kv
if n >= ck then let rec adjpq ck cv pq =
let npq = MinHeap.replaceMin (ck + cv) cv pq
match MinHeap.peekMin npq with
| None -> npq // never happens
| Some(nk, nv) -> if n >= nk then adjpq nk nv npq
else npq
nxtprm (n + inc) (adjpq ck cv pq) q bps
else CIS(n, fun() -> nxtprm (n + inc) pq q bps)
let rec oddprms() = CIS(frstoddprm, fun() ->
nxtprm (frstoddprm + inc) MinHeap.empty (frstoddprm * frstoddprm) (oddprms()))
Seq.unfold (fun (cis: CIS<Prime>) -> Some(cis.v, cis.cont()))
(CIS(frstprm, fun() -> (oddprms())))

The above code is well over five times faster than the previous translated O'Neill version for the given variety of reasons.

Although slightly faster than the Tree Folding code, this latter code is also limited in practical usefulness to about the first one to ten million primes or so.

All of the above codes can be tested in the F# REPL with the following to produce the millionth prime (the "nth" function is zero based):

> primesXXX() |> Seq.nth 999999;;

where primesXXX() is replaced by the given primes generating function to be tested, and which all produce the following output (after a considerable wait in some cases):

Output:
val it : Prime = 15485863u

[edit] Imperative

The following code is written in functional style other than it uses a mutable bit array to sieve the composites:

let primes limit =
let buf = System.Collections.BitArray(int limit + 1, true)
let cull p = { p * p .. p .. limit } |> Seq.iter (fun c -> buf.[int c] <- false)
{ 2u .. uint32 (sqrt (double limit)) } |> Seq.iter (fun c -> if buf.[int c] then cull c)
{ 2u .. limit } |> Seq.map (fun i -> if buf.[int i] then i else 0u) |> Seq.filter ((<>) 0u)
 
[<EntryPoint>]
let main argv =
if argv = null || argv.Length = 0 then failwith "no command line argument for limit!!!"
printfn "%A" (primes (System.UInt32.Parse argv.[0]) |> Seq.length)
0 // return an integer exit code

Substituting the following minor changes to the code for the "primes" function will only deal with the odd prime candidates for a speed up of over a factor of two as well as a reduction of the buffer size by a factor of two:

let primes limit =
let lmtb,lmtbsqrt = (limit - 3u) / 2u, (uint32 (sqrt (double limit)) - 3u) / 2u
let buf = System.Collections.BitArray(int lmtb + 1, true)
let cull i = let p = i + i + 3u in let s = p * (i + 1u) + i in
{ s .. p .. lmtb } |> Seq.iter (fun c -> buf.[int c] <- false)
{ 0u .. lmtbsqrt } |> Seq.iter (fun i -> if buf.[int i] then cull i )
let oddprms = { 0u .. lmtb } |> Seq.map (fun i -> if buf.[int i] then i + i + 3u else 0u)
|> Seq.filter ((<>) 0u)
seq { yield 2u; yield! oddprms }

The following code uses other functional forms for the inner culling loops of the "primes function" to reduce the use of inefficient sequences so as to reduce the execution time by another factor of almost three:

let primes limit =
let lmtb,lmtbsqrt = (limit - 3u) / 2u, (uint32 (sqrt (double limit)) - 3u) / 2u
let buf = System.Collections.BitArray(int lmtb + 1, true)
let rec culltest i = if i <= lmtbsqrt then
let p = i + i + 3u in let s = p * (i + 1u) + i in
let rec cullp c = if c <= lmtb then buf.[int c] <- false; cullp (c + p)
(if buf.[int i] then cullp s); culltest (i + 1u) in culltest 0u
seq {yield 2u; for i = 0u to lmtb do if buf.[int i] then yield i + i + 3u }

Now much of the remaining execution time is just the time to enumerate the primes as can be seen by turning "primes" into a primes counting function by substituting the following for the last line in the above code doing the enumeration; this makes the code run about a further five times faster:

  let rec count i acc =
if i > int lmtb then acc else if buf.[i] then count (i + 1) (acc + 1) else count (i + 1) acc
count 0 1

Since the final enumeration of primes is the main remaining bottleneck, it is worth using a "roll-your-own" enumeration implemented as an object expression so as to save many inefficiencies in the use of the built-in seq computational expression by substituting the following code for the last line of the previous codes, which will decrease the execution time by a factor of over three (instead of almost five for the counting-only version, making it almost as fast):

  let nmrtr() =
let i = ref -2
let rec nxti() = i:=!i + 1;if !i <= int lmtb && not buf.[!i] then nxti() else !i <= int lmtb
let inline curr() = if !i < 0 then (if !i= -1 then 2u else failwith "Enumeration not started!!!")
else let v = uint32 !i in v + v + 3u
{ new System.Collections.Generic.IEnumerator<_> with
member this.Current = curr()
interface System.Collections.IEnumerator with
member this.Current = box (curr())
member this.MoveNext() = if !i< -1 then i:=!i+1;true else nxti()
member this.Reset() = failwith "IEnumerator.Reset() not implemented!!!"a
interface System.IDisposable with
member this.Dispose() = () }
{ new System.Collections.Generic.IEnumerable<_> with
member this.GetEnumerator() = nmrtr()
interface System.Collections.IEnumerable with
member this.GetEnumerator() = nmrtr() :> System.Collections.IEnumerator }

The various optimization techniques shown here can be used "jointly and severally" on any of the basic versions for various trade-offs between code complexity and performance. Not shown here are other techniques of making the sieve faster, including extending wheel factorization to much larger wheels such as 2/3/5/7, pre-culling the arrays, page segmentation, and multi-processing.

[edit] Almost functional Unbounded

the following odds-only implmentations are written in an almost functional style avoiding the use of mutability except for the contents of the data structures uses to hold the state of the and any mutability necessary to implement a "roll-your-own" IEnumberable iterator interface for speed.

Unbounded Dictionary (Mutable Hash Table) Based Sieve

The following code uses the DotNet Dictionary class instead of the above functional Priority Queue to implement the sieve; as average (amortized) hash table access is O(1) rather than O(log n) as for the priority queue, this implementation is slightly faster than the priority queue version for the first million primes and will always be faster for any range above some low range value:

type Prime = uint32
let frstprm = 2u
let frstoddprm = 3u
let inc = 2u
let primesDict() =
let dct = System.Collections.Generic.Dictionary()
let rec nxtprm n q (bps: CIS<Prime>) =
if n >= q then let bp = bps.v in let adv = bp + bp
let nbps = bps.cont() in let nbp = nbps.v
dct.Add(n + adv, adv)
nxtprm (n + inc) (nbp * nbp) nbps
else if dct.ContainsKey(n) then
let adv = dct.[n]
dct.Remove(n) |> ignore
// let mutable nn = n + adv // ugly imperative code
// while dct.ContainsKey(nn) do nn <- nn + adv
// dct.Add(nn, adv)
let rec nxtmt k = // advance to next empty spot
if dct.ContainsKey(k) then nxtmt (k + adv)
else dct.Add(k, adv) in nxtmt (n + adv)
nxtprm (n + inc) q bps
else CIS(n, fun() -> nxtprm (n + inc) q bps)
let rec oddprms() = CIS(frstoddprm, fun() ->
nxtprm (frstoddprm + inc) (frstoddprm * frstoddprm) (oddprms()))
Seq.unfold (fun (cis: CIS<Prime>) -> Some(cis.v, cis.cont()))
(CIS(frstprm, fun() -> (oddprms())))

The above code uses functional forms of code (with the imperative style commented out to show how it could be done imperatively) and also uses a recursive non-sharing secondary source of base primes just as for the Priority Queue version. As for the functional codes, the Primes type can easily be changed to "uint64" for wider range of sieving.

In spite of having true O(n log log n) Sieve of Eratosthenes computational complexity where n is the range of numbers to be sieved, the above code is still not particularly fast due to the time required to compute the hash values and manipulations of the hash table.

Unbounded Page Segmented Mutable Array Sieve

All of the above unbounded implementations including the above Dictionary based version are quite slow due to their large constant factor computational overheads, making them more of an intellectual exercise than something practical, especially when larger sieving ranges are required. The following code implements an unbounded page segmented version of the sieve in not that many more lines of code, yet runs about 25 times faster than the Dictionary version for larger ranges of sieving such as to one billion; it uses functional forms without mutability other than for the contents of the arrays and a reference cell used to implement the "roll-your-own" IEnumerable/IEnumerator interfaces for speed:

let private PGSZBTS = (1 <<< 14) * 8 // sieve buffer size in bits
type private PS = class
val i:int val p:uint64 val cmpsts:uint32[]
new(i,p,c) = { i=i; p=p; cmpsts=c } end
let rec primesPaged(): System.Collections.Generic.IEnumerable<_> =
let lbpse = lazy (primesPaged().GetEnumerator()) // lazy to prevent race
let bpa = ResizeArray() // fills from above sequence as needed
let makePg low =
let nxt = low + (uint64 PGSZBTS <<< 1)
let cmpsts = Array.zeroCreate (PGSZBTS >>> 5)
let inline notprm c = cmpsts.[c >>> 5] &&& (1u <<< c) <> 0u
let rec nxti c = if c < PGSZBTS && notprm c
then nxti (c + 1) else c
let inline mrkc c = let w = c >>> 5
cmpsts.[w] <- cmpsts.[w] ||| (1u <<< c)
let rec cullf i =
if notprm i then cullf (i + 1) else
let p = 3 + i + i in let sqr = p * p
if uint64 sqr < nxt then
let rec cullp c = if c < PGSZBTS then mrkc c; cullp (c + p)
else cullf (i + 1) in cullp ((sqr - 3) >>> 1)
if low <= 3UL then cullf 0 // special culling for the first page
else // cull rest based on a secondary base prime stream
let bpse = lbpse.Force()
if bpa.Count <= 0 then // move past 2 to 3
bpse.MoveNext() |> ignore; bpse.MoveNext() |> ignore
let rec fill np =
if np * np >= nxt then
let bpasz = bpa.Count
let rec cull i =
if i < bpasz then
let p = bpa.[i] in let sqr = p * p in let pi = int p
let strt = if sqr >= low then int (sqr - low) >>> 1
else let r = int (((low - sqr) >>> 1) % p)
if r = 0 then 0 else int p - r
let rec cullp c = if c < PGSZBTS then mrkc c; cullp (c + pi)
cullp strt; cull (i + 1) in cull 0
else bpa.Add(np); bpse.MoveNext() |> ignore
fill bpse.Current
fill bpse.Current // fill pba as necessary and do cull
let ni = nxti 0 in let np = low + uint64 (ni <<< 1)
PS(ni, np, cmpsts)
let nmrtr() =
let ps = ref (PS(0, 0UL, Array.zeroCreate 0))
{ new System.Collections.Generic.IEnumerator<_> with
member this.Current = (!ps).p
interface System.Collections.IEnumerator with
member this.Current = box ((!ps).p)
member this.MoveNext() =
let drps = !ps in let i = drps.i in let p = drps.p
let cmpsts = drps.cmpsts in let lmt = cmpsts.Length <<< 5
if p < 3UL then (if p < 2UL then ps := PS(0, 2UL, cmpsts); true
else ps := makePg 3UL; true) else
let inline notprm c = cmpsts.[c >>> 5] &&& (1u <<< c) <> 0u
let rec nxti c = if c < lmt && notprm c
then nxti (c + 1) else c
let ni = nxti (i + 1) in let np = p + uint64 ((ni - i) <<< 1)
if ni < lmt then ps := PS(ni, np, cmpsts); true
else ps := makePg np; true
member this.Reset() = failwith "IEnumerator.Reset() not implemented!!!"
interface System.IDisposable with
member this.Dispose() = () }
{ new System.Collections.Generic.IEnumerable<_> with
member this.GetEnumerator() = nmrtr()
interface System.Collections.IEnumerable with
member this.GetEnumerator() = nmrtr() :> System.Collections.IEnumerator }

As with all of the efficient unbounded sieves, the above code uses a secondary enumerator of the base primes less than the square root of the currently culled range ("lbpse"), which is this case is a lazy (deffered evaluation) binding so as to avoid a race condition.

The above code is written to output the "uint64" type for very large ranges of primes since there is little computational cost to doing this for this algorithm. As written, the practical range for this sieve is about 16 billion, however, it can be extended to about 10^14 (a week or two of execution time) by setting the "PGSZBTS" constant to the size of the CPU L2 cache rather than the L1 cache (L2 is up to about two Megabytes for modern high end desktop CPU's) at a slight loss of efficiency (a factor of up to two or so) per composite number culling operation due to the slower memory access time.

Even with the custom IEnumerable/IEnumerator interfaces using an object expression (the F# built-in sequence operators are terribly inefficient), the time to enumerate the resulting primes takes longer than the time to actually cull the composite numbers from the sieving arrays. The time to do the actual culling is thus over 50 times faster than done using the Dictionary version. The slowness of enumeration, no matter what further tweaks are done to improve it (each value enumerated will always take function calls and a scan loop that will always take something in the order of 100 CPU clock cycles per value), means that further gains in speed using extreme wheel factorization and multi-processing have little point unless the actual work on the resulting primes is done through use of auxiliary functions not using iteration.

[edit] Forth

: prime? ( n -- ? ) here + c@ 0= ;
: composite! ( n -- ) here + 1 swap c! ;

: sieve ( n -- )
  here over erase
  2
  begin
    2dup dup * >
  while
    dup prime? if
      2dup dup * do
        i composite!
      dup +loop
    then
    1+
  repeat
  drop
  ." Primes: " 2 do i prime? if i . then loop ;

100 sieve

[edit] Fortran

Works with: Fortran version 90 and later
program sieve
 
implicit none
integer, parameter :: i_max = 100
integer :: i
logical, dimension (i_max) :: is_prime
 
is_prime = .true.
is_prime (1) = .false.
do i = 2, int (sqrt (real (i_max)))
if (is_prime (i)) is_prime (i * i : i_max : i) = .false.
end do
do i = 1, i_max
if (is_prime (i)) write (*, '(i0, 1x)', advance = 'no') i
end do
write (*, *)
 
end program sieve

Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

Optimised using a pre-computed wheel based on 2:

program sieve_wheel_2
 
implicit none
integer, parameter :: i_max = 100
integer :: i
logical, dimension (i_max) :: is_prime
 
is_prime = .true.
is_prime (1) = .false.
is_prime (4 : i_max : 2) = .false.
do i = 3, int (sqrt (real (i_max))), 2
if (is_prime (i)) is_prime (i * i : i_max : 2 * i) = .false.
end do
do i = 1, i_max
if (is_prime (i)) write (*, '(i0, 1x)', advance = 'no') i
end do
write (*, *)
 
end program sieve_wheel_2

Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

[edit] GAP

Eratosthenes := function(n)
local a, i, j;
a := ListWithIdenticalEntries(n, true);
if n < 2 then
return [];
else
for i in [2 .. n] do
if a[i] then
j := i*i;
if j > n then
return Filtered([2 .. n], i -> a[i]);
else
while j <= n do
a[j] := false;
j := j + i;
od;
fi;
fi;
od;
fi;
end;
 
Eratosthenes(100);
 
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 ]

[edit] GLBasic

// Sieve of Eratosthenes (find primes)
// GLBasic implementation
 
 
GLOBAL n%, k%, limit%, flags%[]
 
limit = 100 // search primes up to this number
 
DIM flags[limit+1] // GLBasic arrays start at 0
 
FOR n = 2 TO SQR(limit)
IF flags[n] = 0
FOR k = n*n TO limit STEP n
flags[k] = 1
NEXT
ENDIF
NEXT
 
// Display the primes
FOR n = 2 TO limit
IF flags[n] = 0 THEN STDOUT n + ", "
NEXT
 
KEYWAIT
 

[edit] Go

package main
 
import (
"fmt"
)
 
func main() {
const limit = 201 // means sieve numbers < 201
 
// sieve
c := make([]bool, limit) // c for composite. false means prime candidate
c[1] = true // 1 not considered prime
p := 2
for {
// first allowed optimization: outer loop only goes to sqrt(limit)
p2 := p * p
if p2 >= limit {
break
}
// second allowed optimization: inner loop starts at sqr(p)
for i := p2; i < limit; i += p {
c[i] = true // it's a composite
 
}
// scan to get next prime for outer loop
for {
p++
if !c[p] {
break
}
}
}
 
// sieve complete. now print a representation.
for n := 1; n < limit; n++ {
if c[n] {
fmt.Print(" .")
} else {
fmt.Printf("%3d", n)
}
if n%20 == 0 {
fmt.Println("")
}
}
}

Output:

  .  2  3  .  5  .  7  .  .  . 11  . 13  .  .  . 17  . 19  .
  .  . 23  .  .  .  .  . 29  . 31  .  .  .  .  . 37  .  .  .
 41  . 43  .  .  . 47  .  .  .  .  . 53  .  .  .  .  . 59  .
 61  .  .  .  .  . 67  .  .  . 71  . 73  .  .  .  .  . 79  .
  .  . 83  .  .  .  .  . 89  .  .  .  .  .  .  . 97  .  .  .
101  .103  .  .  .107  .109  .  .  .113  .  .  .  .  .  .  .
  .  .  .  .  .  .127  .  .  .131  .  .  .  .  .137  .139  .
  .  .  .  .  .  .  .  .149  .151  .  .  .  .  .157  .  .  .
  .  .163  .  .  .167  .  .  .  .  .173  .  .  .  .  .179  .
181  .  .  .  .  .  .  .  .  .191  .193  .  .  .197  .199  .

A fairly odd sieve tree method:

package main
import "fmt"
 
type xint uint64
type xgen func()(xint)
 
func primes() func()(xint) {
pp, psq := make([]xint, 0), xint(25)
 
var sieve func(xint, xint)xgen
sieve = func(p, n xint) xgen {
m, next := xint(0), xgen(nil)
return func()(r xint) {
if next == nil {
r = n
if r <= psq {
n += p
return
}
 
next = sieve(pp[0] * 2, psq) // chain in
pp = pp[1:]
psq = pp[0] * pp[0]
 
m = next()
}
switch {
case n < m: r, n = n, n + p
case n > m: r, m = m, next()
default: r, n, m = n, n + p, next()
}
return
}
}
 
f := sieve(6, 9)
n, p := f(), xint(0)
 
return func()(xint) {
switch {
case p < 2: p = 2
case p < 3: p = 3
default:
for p += 2; p == n; {
p += 2
if p > n {
n = f()
}
}
pp = append(pp, p)
}
return p
}
}
 
 
func main() {
for i, p := 0, primes(); i < 100000; i++ {
fmt.Println(p())
}
}

See also the concurrent prime sieve example in the "Try Go" window at http://golang.org/

[edit] Groovy

This solution uses a BitSet for compactness and speed, but in Groovy, BitSet has full List semantics. It also uses both the "square root of the boundary" shortcut and the "square of the prime" shortcut.

def sievePrimes = { bound -> 
def isPrime = new BitSet(bound)
isPrime[0..1] = false
isPrime[2..bound] = true
(2..(Math.sqrt(bound))).each { pc ->
if (isPrime[pc]) {
((pc**2)..bound).step(pc) { isPrime[it] = false }
}
}
(0..bound).findAll { isPrime[it] }
}

Test:

println sievePrimes(100)

Output:

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

[edit] GW-BASIC

10  INPUT "ENTER NUMBER TO SEARCH TO: ";LIMIT
20 DIM FLAGS(LIMIT)
30 FOR N = 2 TO SQR (LIMIT)
40 IF FLAGS(N) < > 0 GOTO 80
50 FOR K = N * N TO LIMIT STEP N
60 FLAGS(K) = 1
70 NEXT K
80 NEXT N
90 REM DISPLAY THE PRIMES
100 FOR N = 2 TO LIMIT
110 IF FLAGS(N) = 0 THEN PRINT N;", ";
120 NEXT N

[edit] Haskell

[edit] Mutable unboxed arrays, odds only

Mutable array of unboxed Bools indexed by Ints, representing odds only:

import Control.Monad (forM_, when)
import Control.Monad.ST
import Data.Array.ST
import Data.Array.Unboxed
 
sieveUO :: Int -> UArray Int Bool
sieveUO top = runSTUArray $ do
let m = (top-1) `div` 2
r = floor . sqrt $ fromIntegral top + 1
sieve <- newArray (1,m) True -- :: ST s (STUArray s Int Bool)
forM_ [1..r `div` 2] $ \i -> do -- prime(i) = 2i+1
isPrime <- readArray sieve i -- ((2i+1)^2-1)`div`2 = 2i(i+1)
when isPrime $ do
forM_ [2*i*(i+1), 2*i*(i+2)+1..m] $ \j -> do
writeArray sieve j False
return sieve
 
primesToUO :: Int -> [Int]
primesToUO top | top > 1 = 2 : [2*i + 1 | (i,True) <- assocs $ sieveUO top]
| otherwise = []

This represents odds only in the array. Empirical orders of growth is ~ n1.2 in n primes produced, and improving for bigger n‍ ‍s. Memory consumption is low (array seems to be packed) and growing about linearly with n. Can further be significantly sped up by re-writing the forM_ loops with direct recursion, and using unsafeRead and unsafeWrite operations.

[edit] Immutable arrays

This is the most straightforward:

import Data.Array.Unboxed
 
primesToA m = sieve 3 (array (3,m) [(i,odd i) | i<-[3..m]] :: UArray Int Bool)
where
sieve p a
| p*p > m = 2 : [i | (i,True) <- assocs a]
| a!p = sieve (p+2) $ a//[(i,False) | i <- [p*p, p*p+2*p..m]]
| otherwise = sieve (p+2) a

Its performance sharply depends on compiler optimizations. Compiled with -O2 flag in the presence of the explicit type signature, it is very fast in producing first few million primes. (//) is an array update operator.

[edit] Immutable arrays, by segments

Works by segments between consecutive primes' squares. Should be the fastest of non-monadic code:

import Data.Array.Unboxed
 
primesSA = 2 : prs ()
where
prs () = 3 : sieve 3 [] (prs ())
sieve x fs (p:ps) = [i*2 + x | (i,True) <- assocs a]
++ sieve (p*p) fs2 ps
where
q = (p*p-x)`div`2
fs2 = (p,0) : [(s, rem (y-q) s) | (s,y) <- fs]
a :: UArray Int Bool
a = accumArray (\ b c -> False) True (1,q-1)
[(i,()) | (s,y) <- fs, i <- [y+s, y+s+s..q]]

[edit] Basic list-based sieve

Straightforward implementation of the sieve of Eratosthenes in its original bounded form. This finds primes in gaps between the composites, and composites as an enumeration of each prime's multiples.

primesTo m = 2 : eratos [3,5..m] where
eratos (p : xs) | p*p>m = p : xs
| otherwise = p : eratos (xs `minus` [p*p, p*p+2*p..m])
 
minus a@(x:xs) b@(y:ys) = case compare x y of
LT -> x : minus xs b
EQ -> minus xs ys
GT -> minus a ys
minus a b = a

Its time complexity is similar to that of optimal trial division because of limitations of Haskell linked lists, where (minus a b) takes time proportional to length(union a b) and not (length b), as achieved in imperative setting with direct-access memory. Uses ordered list representation of sets.

This is reasonably useful up to ranges of fifteen million or about the first million primes.

[edit] Unbounded list based sieve

Unbounded, "naive", too eager to subtract (see above for the definition of minus):

primesE  = sieve [2..] 
where
sieve (p:xs) = p : sieve (minus xs [p,p+p..])

This is slow, with complexity increasing as a square law or worse so that it is only moderately useful up to the first 10,000 primes or so.

The number of active streams can be limited to what's strictly necessary by postponement until after the square of a prime is seen:

primesEQ = 2 : sieve [3..] 4 primesEQ
where
sieve (x:xs) q (p:t)
| x < q = x : sieve xs q (p:t)
| otherwise = sieve (minus xs [q, q+p..]) (head t^2) t

The above code is useful to a range of the first million primes or so.

The basic gradually-deepening left-leaning (((a-b)-c)- ... ) workflow above can be rearranged into the right-leaning (a-(b+(c+ ... ))). This is the idea behind Richard Bird's unbounded code presented in M. O'Neill's article.

primes = _Y $ ((2:) . minus [3..] 
. foldr (\x-> (x*x :) . union [x*x+x, x*x+2*x..]) [])
 
_Y g = g (_Y g) -- = g . g . g . ... non-sharing multistage fixpoint combinator
-- = let x = g x in g x -- = g (fix g) two-stage fixpoint combinator
-- = let x = g x in x -- = fix g sharing fixpoint combinator
 
union a@(x:xs) b@(y:ys) = case compare x y of
LT -> x : union xs b
EQ -> x : union xs ys
GT -> y : union a ys

Using _Y is meant to guarantee the separate supply of primes to be independently calculated, recursively, instead of the same one being reused, corecursively; thus the memory footprint is drastically reduced.

The above code is also useful to a range of the first million primes or so. The code can be further optimized by fusing minus [3..] into one function, preventing a space leak with the newer GHC versions, getting the function gaps defined below.

[edit] List-based tree-merging incremental sieve

Linear merging structure can further be replaced with an indefinitely deepening to the right tree-like structure, (a-(b+((c+d)+( ((e+f)+(g+h)) + ... )))).

This merges primes' multiples streams in a tree-like fashion, achieving theoretical time complexity which is only a log n factor above the optimal n log n log (log n), for n primes produced. Indeed, empirically it runs at about ~ n1.2 (for producing first few million primes), similarly to priority-queue–based version of M. O'Neill's, and with very low space complexity too (not counting the produced sequence of course):

primes :: [Int]   
primes = 2 : _Y ((3 :) . gaps 5 . _U . map(\p-> [p*p, p*p+2*p..]))
 
gaps k s@(x:xs) | k < x = k : gaps (k+2) s -- ~= ([k,k+2..] \\ s)
| otherwise = gaps (k+2) xs -- when null(s\\[k,k+2..])
 
_U ((x:xs):t) = x : (union xs . _U . pairs) t -- ~= nub . sort . concat
where
pairs (xs:ys:t) = union xs ys : pairs t

Here's the test entry on Ideone.com, a comparison with more versions, and a similar code with wheel optimization.

[edit] Priority Queue based incremental sieve

The above work is derived from the Epilogue of the Melissa E. O'Neill paper which is much referenced with respect to incremental functional sieves; however, that paper is now dated and her comments comparing list based sieves to her original work leading up to a Priority Queue based implementation is no longer current given more recent work such as the above Tree Merging version. Accordingly, a modern "odd's-only" Priority Queue version is developed here for more current comparisons between the above list based incremental sieves and a continuation of O'Neill's work.

In order to implement a Priority Queue version with Haskell, an efficient Priority Queue, which is not part of the standard Haskell libraries is required. A Min Heap implementation is likely best suited for this task in providing the most efficient frequently used peeks of the next item in the queue and replacement of the first item in the queue (not using a "pop" followed by a "push) with "pop" operations then not used at all, and "push" operations used relatively infrequently. Judging by O'Neill's use of an efficient "deleteMinAndInsert" operation which she states "(We provide deleteMinAndInsert becausea heap-based implementation can support this operation with considerably less rearrangement than a deleteMin followed by an insert.)", which statement is true for a Min Heap Priority Queue and not others, and her reference to a priority queue by (Paulson, 1996), the queue she used is likely the one as provided as a simple true functional Min Heap implementation on RosettaCode, from which the essential functions are reproduced here:

data PriorityQ k v = Mt
| Br !k v !(PriorityQ k v) !(PriorityQ k v)
deriving (Eq, Ord, Read, Show)
 
emptyPQ :: PriorityQ k v
emptyPQ = Mt
 
peekMinPQ :: PriorityQ k v -> Maybe (k, v)
peekMinPQ Mt = Nothing
peekMinPQ (Br k v _ _) = Just (k, v)
 
pushPQ :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v
pushPQ wk wv Mt = Br wk wv Mt Mt
pushPQ wk wv (Br vk vv pl pr)
| wk <= vk = Br wk wv (pushPQ vk vv pr) pl
| otherwise = Br vk vv (pushPQ wk wv pr) pl
 
siftdown :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v -> PriorityQ k v
siftdown wk wv Mt _ = Br wk wv Mt Mt
siftdown wk wv (pl @ (Br vk vv _ _)) Mt
| wk <= vk = Br wk wv pl Mt
| otherwise = Br vk vv (Br wk wv Mt Mt) Mt
siftdown wk wv (pl @ (Br vkl vvl pll plr)) (pr @ (Br vkr vvr prl prr))
| wk <= vkl && wk <= vkr = Br wk wv pl pr
| vkl <= vkr = Br vkl vvl (siftdown wk wv pll plr) pr
| otherwise = Br vkr vvr pl (siftdown wk wv prl prr)
 
replaceMinPQ :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v
replaceMinPQ wk wv Mt = Mt
replaceMinPQ wk wv (Br _ _ pl pr) = siftdown wk wv pl pr
 

The "peekMin" function retrieves both of the key and value in a tuple so processing is required to access whichever is required for further processing. As well, the output of the peekMin function is a Maybe with the case of an empty queue providing a Nothing output.

The following code is O'Neill's original odds-only code (without wheel factorization) from her paper slightly adjusted as per the requirements of this Min Heap implementation as laid out above; note the `seq` adjustments to the "adjust" function to make the evaluation of the entry tuple more strict for better efficiency:

 
-- (c) 2006-2007 Melissa O'Neill. Code may be used freely so long as
-- this copyright message is retained and changed versions of the file
-- are clearly marked.
-- the only changes are the names of the called PQ functions and the
-- included processing for the result of the peek function being a maybe tuple.
 
primesPQ() = 2 : sieve [3,5..]
where
sieve [] = []
sieve (x:xs) = x : sieve' xs (insertprime x xs emptyPQ)
where
insertprime p xs table = pushPQ (p*p) (map (* p) xs) table
sieve'
[] table = []
sieve' (x:xs) table
| nextComposite <= x = sieve'
xs (adjust table)
| otherwise = x : sieve' xs (insertprime x xs table)
where
nextComposite = case peekMinPQ table of
Just (c, _) -> c
adjust table
| n <= x = adjust (replaceMinPQ n'
ns table)
| otherwise = table
where (n, n':ns) = case peekMinPQ table of
Just tpl -> tpl

The above code is almost four times slower than the version of the Tree Merging sieve above for the first million primes although it is about the same speed as the original Richard Bird sieve with the "odds-only" adaptation as above. It is slow and uses a huge amount of memory for primarily one reason: over eagerness in adding prime composite streams to the queue, which are added as the primes are listed rather than when they are required as the output primes stream reaches the square of a given base prime; this over eagerness also means that the processed numbers must have a large range in order to not overflow when squared (as in the default Integer = infinite precision integers as used here and by O'Neill, but Int64's or Word64's would give a practical range) which processing of wide range numbers adds processing and memory requirement overhead. Although O'Neill's code is elegant, it also loses some efficiency due to the extensive use of lazy list processing, not all of which is required even for a wheel factorization implementation.

The following code is adjusted to reduce the amount of lazy list processing and to add a secondary base primes stream (or a succession of streams when the combinator is used) so as to overcome the above problems and reduce memory consumption to only that required for the primes below the square root of the currently sieved number; using this means that 32-bit Int's are sufficient for a reasonable range and memory requirements become relatively negligible:

primesPQx :: () -> [Int]
primesPQx() = 2 : _Y ((3 :) . sieve 5 emptyPQ 9) -- initBasePrms
where
_Y g = g (_Y g) -- non-sharing multi-stage fixpoint combinator OR
-- initBasePrms = 3 : sieve 5 emptyPQ 9 initBasePrms -- single stage
insertprime p table = let adv = 2 * p in let nv = p * p + adv in
nv `seq` pushPQ nv adv table
sieve n table q bps@(bp:bps')
| n >= q = let nbp = head bps'
in
sieve (n + 2) (insertprime bp table) (nbp * nbp) bps'
| n >= nextComposite = sieve (n + 2) (adjust table) q bps
| otherwise = n : sieve (n + 2) table q bps
where
nextComposite = case peekMinPQ table of
Nothing -> q -- at beginning when queue empty
Just (c, _) -> c
adjust table
| c <= n = let nc = c + adv in
nc `seq` adjust (replaceMinPQ nc adv table)
| otherwise = table
where (c, adv) = case peekMinPQ table of
Just ct -> ct

The above code is over five times faster than the previous (O'Neill) Priority Queue code and about half again faster than the Tree Merging code for a range of a million primes, and will always be faster as the Min Heap is slightly more efficient than Tree Merging due to better tree balancing.

All of these codes including the list based ones would enjoy about the same constant factor improvement of up to about four times the speed with the application of maximum wheel factorization.

[edit] Page Segmented Sieve using a mutable unboxed array

All of the above unbounded sieves are quite limited in practical sieving range due to the large constant factor overheads in computation, making them mostly just interesting intellectual exercises other than for small ranges of about the first million to ten million primes; the following "odds-only page-segmented version using (bit-packed internally) mutable unboxed arrays is about 50 times faster than the fastest of the above algorithms for ranges of about that and higher, making it practical for the first several hundred million primes:

import Data.Bits
import Data.Array.Base
import Control.Monad.ST
import Data.Array.ST (runSTUArray, STUArray(..))
 
type PrimeType = Int
szPGBTS = (2^14) * 8 :: PrimeType -- CPU L1 cache in bits
 
primesPaged :: () -> [PrimeType]
primesPaged() = 2 : _Y (listPagePrms . pagesFrom 0) where
_Y g = g (_Y g) -- non-sharing multi-stage fixpoint combinator
listPagePrms (hdpg @ (UArray lowi _ rng _) : tlpgs) =
let loop i = if i >= rng then listPagePrms tlpgs
else if unsafeAt hdpg i then loop (i + 1)
else let ii = lowi + fromIntegral i in
case 3 + ii + ii of
p -> p `seq` p : loop (i + 1) in loop 0
makePg lowi bps = runSTUArray $ do
let limi = lowi + szPGBTS - 1
let nxt = 3 + limi + limi -- last candidate in range
cmpsts <- newArray (lowi, limi) False
let pbts = fromIntegral szPGBTS
let cull (p:ps) =
let sqr = p * p in
if sqr > nxt then return cmpsts
else let pi = fromIntegral p in
let cullp c = if c > pbts then return ()
else do
unsafeWrite cmpsts c True
cullp (c + pi) in
let a = (sqr - 3) `shiftR` 1 in
let s = if a >= lowi then fromIntegral (a - lowi)
else let r = fromIntegral ((lowi - a) `rem` p) in
if r == 0 then 0 else pi - r in
do { cullp s; cull ps}
if lowi == 0 then do
pg0 <- unsafeFreezeSTUArray cmpsts
cull $ listPagePrms [pg0]
else cull bps
pagesFrom lowi bps =
let cf lwi = case makePg lwi bps of
pg -> pg `seq` pg : cf (lwi + szPGBTS) in cf lowi

The above code is currently implemented to use "Int" as the prime type but one can change the "PrimeType" to "Int64" (importing Data.Int) or "Word64" (importing Data.Word) to extend the range to its maximum practical range of above 10^14 or so. Note that for larger ranges that one will want to set the "szPGBTS" to something close to the CPU L2 or even L3 cache size (up to 8 Megabytes = 2^23 for an Intel i7) for a slight cost in speed (about a factor of 1.5) but so that it still computes fairly efficiently as to memory access up to those large ranges. It would be quite easy to modify the above code to make the page array size automatically increase in size with increasing range.

The above code takes only a few tens of milliseconds to compute the first million primes and a few seconds to calculate the first 50 million primes, with over half of those times expended in just enumerating the result lazy list, with even worse times when using 64-bit list processing (especially with 32-bit versions of GHC). A further improvement to reduce the computational cost of repeated list processing across the base pages for every page segment would be to store the required base primes (or base prime gaps) in an array that gets extended in size by factors of two (by copying the old array to the new extended array) as the number of base primes increases; in that way the scans across base primes per page segment would just be array accesses which are much faster than list enumeration.

Unlike many other other unbounded examples, this algorithm has the true Sieve of Eratosthenes computational time complexity of O(n log log n) where n is the sieving range with no extra "log n" factor while having a very low computational time cost per composite number cull of less than ten CPU clock cycles per cull (well under as in under 4 clock cycles for the Intel i7 using a page buffer size of the CPU L1 cache size).

There are other ways to make the algorithm faster including high degrees of wheel factorization, which can reduce the number of composite culling operations by a factor of about four for practical ranges, and multi-processing which can reduce the computation time proportionally to the number of available independent CPU cores, but there is little point to these optimizations as long as the lazy list enumeration is the bottleneck as it is starting to be in the above code. To take advantage of those optimizations, functions need to be provided that can compute the desired results without using list processing.

For ranges above about 10^14 where culling spans begin to exceed even an expanded size page array, other techniques need to be adapted such as the use of a "bucket sieve" which tracks the next page that larger base prime culling sequences will "hit" to avoid redundant (and time expensive) start address calculations for base primes that don't "hit" the current page.

However, even with the above code and its limitations for large sieving ranges, the speeds will never come close to as slow as the other "incremental" sieve algorithms, as the time will never exceed about 100 CPU clock cycles per composite number cull, where the fastest of those other algorithms takes many hundreds of CPU clock cycles per cull.

[edit] APL-style

Rolling set subtraction over the rolling element-wise addition on integers. Basic:

zipWith (flip (!!)) [0..]
. scanl1 minus . scanl1 (zipWith (+)) $ repeat [2..]

A bit optimized:

tail . concatMap fst
. (\(x:xs)-> scanl (\(_,a) b-> span (< head b) $ minus a b)
([],x) xs)
. scanl1 (zipWith (+) . tail) $ tails [1..]

An illustration:

> mapM_ (print . take 15) $ take 10 $ scanl1 (zipWith(+)) $ repeat [2..]
[ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
[ 4, 6, 8,10,12,14,16,18, 20, 22, 24, 26, 28, 30]
[ 6, 9,12,15,18,21,24,27, 30, 33, 36, 39, 42, 45]
[ 8,12,16,20,24,28,32,36, 40, 44, 48, 52, 56, 60]
[10,15,20,25,30,35,40,45, 50, 55, 60, 65, 70, 75]
[12,18,24,30,36,42,48,54, 60, 66, 72, 78, 84, 90]
[14,21,28,35,42,49,56,63, 70, 77, 84, 91, 98,105]
[16,24,32,40,48,56,64,72, 80, 88, 96,104,112,120]
[18,27,36,45,54,63,72,81, 90, 99,108,117,126,135]
[20,30,40,50,60,70,80,90,100,110,120,130,140,150]
 
> mapM_ (print . take 15) $ take 10 $ scanl1 (zipWith(+) . tail) $ tails [1..]
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
[ 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32]
[ 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51]
[ 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72]
[ 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95]
[ 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96,102,108,114,120]
[ 49, 56, 63, 70, 77, 84, 91, 98,105,112,119,126,133,140,147]
[ 64, 72, 80, 88, 96,104,112,120,128,136,144,152,160,168,176]
[ 81, 90, 99,108,117,126,135,144,153,162,171,180,189,198,207]
[100,110,120,130,140,150,160,170,180,190,200,210,220,230,240]

[edit] HicEst

REAL :: N=100,  sieve(N)
 
sieve = $ > 1 ! = 0 1 1 1 1 ...
DO i = 1, N^0.5
IF( sieve(i) ) THEN
DO j = i^2, N, i
sieve(j) = 0
ENDDO
ENDIF
ENDDO
 
DO i = 1, N
IF( sieve(i) ) WRITE() i
ENDDO

[edit] Icon and Unicon

 procedure main()
sieve(100)
end
 
procedure sieve(n)
local p,i,j
p:=list(n, 1)
every i:=2 to sqrt(n) & j:= i+i to n by i & p[i] == 1
do p[j] := 0
every write(i:=2 to n & p[i] == 1 & i)
end

Alternatively using sets

 procedure main()
sieve(100)
end
 
procedure sieve(n)
primes := set()
every insert(primes,1 to n)
every member(primes,i := 2 to n) do
every delete(primes,i + i to n by i)
delete(primes,1)
every write(!sort(primes))
end

[edit] J

Generally, this task should be accomplished in J using i.&.(p:inv) . Here we take an approach that's more comparable with the other examples on this page.

This problem is a classic example of how J can be used to represent mathematical concepts.

J uses x|y (residue) to represent the operation of finding the remainder during integer division of y divided by x

   10|13
3

And x|/y (table) gives us a table with all possibilities from x and all possibilities from y.

   2 3 4 |/ 2 3 4
0 1 0
2 0 1
2 3 0

Meanwhile, |/~y (reflex) copies the right argument and uses it as the left argment.

   |/~ 0 1 2 3 4
0 1 2 3 4
0 0 0 0 0
0 1 0 1 0
0 1 2 0 1
0 1 2 3 0

(Bigger examples might make the patterns more obvious but they also take up more space.)

By the way, we can ask J to count out the first N integers for us using i. (integers):

   i. 5
0 1 2 3 4

Anyways, the 0s in that last table represent the Sieve of Eratosthenes (in a symbolic or mathematical sense), and we can use = (equal) to find them.

   0=|/~ i.5
1 0 0 0 0
1 1 1 1 1
1 0 1 0 1
1 0 0 1 0
1 0 0 0 1

Now all we need to do is add them up, using / (insert) in its single argument role to insert + between each row of that last table.

   +/0=|/~ i.5
5 1 2 2 3

The sieve wants the cases where we have two divisors:

   2=+/0=|/~ i.5
0 0 1 1 0

And we just want to know the positions of the 1s in that list, which we can find using I. (indices):

   I.2=+/0=|/~ i.5
2 3
I.2=+/0=|/~ i.100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

And we might want to express this sentence as a definition of a word that lets us use it with an arbitrary argument. There are a variety of ways of doing this. For example:

sieve0=: verb def 'I.2=+/0=|/~ i.y'

Of course, we can also express this in an even more elaborate fashion. The elaboration makes more efficient use of resources for large arguments, at the expense of less efficient use of resources for small arguments:

sieve1=: 3 : 0
m=. <.%:y
z=. $0
b=. y{.1
while. m>:j=. 1+b i. 0 do.
b=. b+.y$(-j){.1
z=. z,j
end.
z,1+I.-.b
)

"Wheels" may be implemented as follows:

sieve2=: 3 : 0
m=. <.%:y
z=. y (>:#]) 2 3 5 7
b=. 1,}.y$+./(*/z)$&>(-z){.&.>1
while. m>:j=. 1+b i. 0 do.
b=. b+.y$(-j){.1
z=. z,j
end.
z,1+I.-.b
)

The use of 2 3 5 7 as wheels provides a 20% time improvement for n=1000 and 2% for n=1e6 but note that sieve2 is still 25 times slower than i.&.(p:inv) for n=1e6. Then again, the value of the sieve of eratosthenes was not efficiency but simplicity. So perhaps we should ignore resource consumption issues and instead focus on intermediate results for reasonably sized example problems?

   0=|/~ i.8
1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
1 0 1 0 1 0 1 0
1 0 0 1 0 0 1 0
1 0 0 0 1 0 0 0
1 0 0 0 0 1 0 0
1 0 0 0 0 0 1 0
1 0 0 0 0 0 0 1

Columns with two "1" values correspond to prime numbers.

[edit] Java

Works with: Java version 1.5+
import java.util.LinkedList;
 
public class Sieve{
public static LinkedList<Integer> sieve(int n){
if(n < 2) return new LinkedList<Integer>();
LinkedList<Integer> primes = new LinkedList<Integer>();
LinkedList<Integer> nums = new LinkedList<Integer>();
 
for(int i = 2;i <= n;i++){ //unoptimized
nums.add(i);
}
 
while(nums.size() > 0){
int nextPrime = nums.remove();
for(int i = nextPrime * nextPrime;i <= n;i += nextPrime){
nums.removeFirstOccurrence(i);
}
primes.add(nextPrime);
}
return primes;
}
}

To optimize by testing only odd numbers, replace the loop marked "unoptimized" with these lines:

nums.add(2);
for(int i = 3;i <= n;i += 2){
nums.add(i);
}

Version using a BitSet:

import java.util.LinkedList;
import java.util.BitSet;
 
public class Sieve{
public static LinkedList<Integer> sieve(int n){
LinkedList<Integer> primes = new LinkedList<Integer>();
BitSet nonPrimes = new BitSet(n+1);
 
for (int p = 2; p <= n ; p = nonPrimes.nextClearBit(p+1)) {
for (int i = p * p; i <= n; i += p)
nonPrimes.set(i);
primes.add(p);
}
return primes;
}
}

[edit] Infinite iterator

An iterator that will generate primes indefinitely (perhaps until it runs out of memory), but very slowly.

Translation of: Python
Works with: Java version 1.5+
import java.util.Iterator;
import java.util.PriorityQueue;
import java.math.BigInteger;
 
// generates all prime numbers
public class InfiniteSieve implements Iterator<BigInteger> {
 
private static class NonPrimeSequence implements Comparable<NonPrimeSequence> {
BigInteger currentMultiple;
BigInteger prime;
 
public NonPrimeSequence(BigInteger p) {
prime = p;
currentMultiple = p.multiply(p); // start at square of prime
}
@Override public int compareTo(NonPrimeSequence other) {
// sorted by value of current multiple
return currentMultiple.compareTo(other.currentMultiple);
}
}
 
private BigInteger i = BigInteger.valueOf(2);
// priority queue of the sequences of non-primes
// the priority queue allows us to get the "next" non-prime quickly
final PriorityQueue<NonPrimeSequence> nonprimes = new PriorityQueue<NonPrimeSequence>();
 
@Override public boolean hasNext() { return true; }
@Override public BigInteger next() {
// skip non-prime numbers
for ( ; !nonprimes.isEmpty() && i.equals(nonprimes.peek().currentMultiple); i = i.add(BigInteger.ONE)) {
// for each sequence that generates this number,
// have it go to the next number (simply add the prime)
// and re-position it in the priority queue
while (nonprimes.peek().currentMultiple.equals(i)) {
NonPrimeSequence x = nonprimes.poll();
x.currentMultiple = x.currentMultiple.add(x.prime);
nonprimes.offer(x);
}
}
// prime
// insert a NonPrimeSequence object into the priority queue
nonprimes.offer(new NonPrimeSequence(i));
BigInteger result = i;
i = i.add(BigInteger.ONE);
return result;
}
 
public static void main(String[] args) {
Iterator<BigInteger> sieve = new InfiniteSieve();
for (int i = 0; i < 25; i++) {
System.out.println(sieve.next());
}
}
}
Output:
2
3
5
7
11
13
17
19

[edit] Infinite iterator with a faster algorithm (sieves odds-only)

The adding of each discovered prime's incremental step information to the mapping should be postponed until the candidate number reaches the primes square, as it is useless before that point. This drastically reduces the space complexity from O(n/log(n)) to O(sqrt(n/log(n))), in n primes produced, and also lowers the run time complexity due to the use of the hash table based HashMap, which is much more efficient for large ranges.

Translation of: Python
Works with: Java version 1.5+
import java.util.Iterator;
import java.util.HashMap;
 
// generates all prime numbers up to about 10 ^ 19 if one can wait 1000's of years or so...
public class SoEInfHashMap implements Iterator<Long> {
 
long candidate = 2;
Iterator<Long> baseprimes = null;
long basep = 3;
long basepsqr = 9;
// HashMap of the sequences of non-primes
// the hash map allows us to get the "next" non-prime reasonably quickly
// but further allows re-insertions to take amortized constant time
final HashMap<Long,Long> nonprimes = new HashMap<>();
 
@Override public boolean hasNext() { return true; }
@Override public Long next() {
// do the initial primes separately to initialize the base primes sequence
if (this.candidate <= 5L) if (this.candidate++ == 2L) return 2L; else {
this.candidate++; if (this.candidate == 5L) return 3L; else {
this.baseprimes = new SoEInfHashMap();
this.baseprimes.next(); this.baseprimes.next(); // throw away 2 and 3
return 5L;
} }
// skip non-prime numbers including squares of next base prime
for ( ; this.candidate >= this.basepsqr || //equals nextbase squared => not prime
nonprimes.containsKey(this.candidate); candidate += 2) {
// insert a square root prime sequence into hash map if to limit
if (candidate >= basepsqr) { // if square of base prime, always equal
long adv = this.basep << 1;
nonprimes.put(this.basep * this.basep + adv, adv);
this.basep = this.baseprimes.next();
this.basepsqr = this.basep * this.basep;
}
// else for each sequence that generates this number,
// have it go to the next number (simply add the advance)
// and re-position it in the hash map at an emply slot
else {
long adv = nonprimes.remove(this.candidate);
long nxt = this.candidate + adv;
while (this.nonprimes.containsKey(nxt)) nxt += adv; //unique keys
this.nonprimes.put(nxt, adv);
}
}
// prime
long tmp = candidate; this.candidate += 2; return tmp;
}
 
public static void main(String[] args) {
int n = 100000000;
long strt = System.currentTimeMillis();
SoEInfHashMap sieve = new SoEInfHashMap();
int count = 0;
while (sieve.next() <= n) count++;
long elpsd = System.currentTimeMillis() - strt;
System.out.println("Found " + count + " primes up to " + n + " in " + elpsd + " milliseconds.");
}
 
}
Output:
Found 5761455 primes up to 100000000 in 4297 milliseconds.

[edit] Infinite iterator with a very fast page segmentation algorithm (sieves odds-only)

Although somewhat faster than the previous infinite iterator version, the above code is still over 10 times slower than an infinite iterator based on array paged segmentation as in the following code, where the time to enumerate/iterate over the found primes (common to all the iterators) is now about half of the total execution time:

Translation of: JavaScript
Works with: Java version 1.5+
import java.util.Iterator;
import java.util.ArrayList;
 
// generates all prime numbers up to about 10 ^ 19 if one can wait 100's of years or so...
// practical range is about 10^14 in a week or so...
public class SoEPagedOdds implements Iterator<Long> {
private final int BFSZ = 1 << 16;
private final int BFBTS = BFSZ * 32;
private final int BFRNG = BFBTS * 2;
private long bi = -1;
private long lowi = 0;
private final ArrayList<Integer> bpa = new ArrayList<>();
private Iterator<Long> bps;
private final int[] buf = new int[BFSZ];
 
@Override public boolean hasNext() { return true; }
@Override public Long next() {
if (this.bi < 1) {
if (this.bi < 0) {
this.bi = 0;
return 2L;
}
//this.bi muxt be 0
long nxt = 3 + (this.lowi << 1) + BFRNG;
if (this.lowi <= 0) { // special culling for first page as no base primes yet:
for (int i = 0, p = 3, sqr = 9; sqr < nxt; i++, p += 2, sqr = p * p)
if ((this.buf[i >>> 5] & (1 << (i & 31))) == 0)
for (int j = (sqr - 3) >> 1; j < BFBTS; j += p)
this.buf[j >>> 5] |= 1 << (j & 31);
}
else { // after the first page:
for (int i = 0; i < this.buf.length; i++)
this.buf[i] = 0; // clear the sieve buffer
if (this.bpa.isEmpty()) { // if this is the first page after the zero one:
this.bps = new SoEPagedOdds(); // initialize separate base primes stream:
this.bps.next(); // advance past the only even prime of two
this.bpa.add(this.bps.next().intValue()); // get the next prime (3 in this case)
}
// get enough base primes for the page range...
for (long p = this.bpa.get(this.bpa.size() - 1), sqr = p * p; sqr < nxt;
p = this.bps.next(), this.bpa.add((int)p), sqr = p * p) ;
for (int i = 0; i < this.bpa.size() - 1; i++) {
long p = this.bpa.get(i);
long s = (p * p - 3) >>> 1;
if (s >= this.lowi) // adjust start index based on page lower limit...
s -= this.lowi;
else {
long r = (this.lowi - s) % p;
s = (r != 0) ? p - r : 0;
}
for (int j = (int)s; j < BFBTS; j += p)
this.buf[j >>> 5] |= 1 << (j & 31);
}
}
}
while ((this.bi < BFBTS) &&
((this.buf[(int)this.bi >>> 5] & (1 << ((int)this.bi & 31))) != 0))
this.bi++; // find next marker still with prime status
if (this.bi < BFBTS) // within buffer: output computed prime
return 3 + ((this.lowi + this.bi++) << 1);
else { // beyond buffer range: advance buffer
this.bi = 0;
this.lowi += BFBTS;
return this.next(); // and recursively loop
}
}
 
public static void main(String[] args) {
long n = 1000000000;
long strt = System.currentTimeMillis();
Iterator<Long> gen = new SoEPagedOdds();
int count = 0;
while (gen.next() <= n) count++;
long elpsd = System.currentTimeMillis() - strt;
System.out.println("Found " + count + " primes up to " + n + " in " + elpsd + " milliseconds.");
}
 
}
Output:
Found 50847534 primes up to 1000000000 in 3201 milliseconds.

[edit] JavaScript

function eratosthenes(limit) {
var primes = [];
if (limit >= 2) {
var sqrtlmt = Math.sqrt(limit) - 2;
var nums = new Array(); // start with an empty Array...
for (var i = 2; i <= limit; i++) // and
nums.push(i); // only initialize the Array once...
for (var i = 0; i <= sqrtlmt; i++) {
var p = nums[i]
if (p)
for (var j = p * p - 2; j < nums.length; j += p)
nums[j] = 0;
}
for (var i = 0; i < nums.length; i++) {
var p = nums[i];
if (p)
primes.push(p);
}
}
return primes;
}
 
var primes = eratosthenes(100);
 
if (typeof print == "undefined")
print = (typeof WScript != "undefined") ? WScript.Echo : alert;
print(primes);

outputs:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

Substituting the following code for the function for an odds-only algorithm using bit packing for the array produces code that is many times faster than the above:

function eratosthenes(limit) {
var prms = [];
if (limit >= 2) prms = [2];
if (limit >= 3) {
var sqrtlmt = (Math.sqrt(limit) - 3) >> 1;
var lmt = (limit - 3) >> 1;
var bfsz = (lmt >> 5) + 1
var buf = [];
for (var i = 0; i < bfsz; i++)
buf.push(0);
for (var i = 0; i <= sqrtlmt; i++)
if ((buf[i >> 5] & (1 << (i & 31))) == 0) {
var p = i + i + 3;
for (var j = (p * p - 3) >> 1; j <= lmt; j += p)
buf[j >> 5] |= 1 << (j & 31);
}
for (var i = 0; i <= lmt; i++)
if ((buf[i >> 5] & (1 << (i & 31))) == 0)
prms.push(i + i + 3);
}
return prms;
}

While the above code is quite fast especially using an efficient JavaScript engine such as Google Chrome's V8, it isn't as elegant as it could be using the features of the new EcmaScript6 specification when it comes out about the end of 2014 and when JavaScript engines including those of browsers implement that standard in that we might choose to implement an incremental algorithm iterators or generators similar to as implemented in Python or F# (yield). Meanwhile, we can emulate some of those features by using a simulation of an iterator class (which is easier than using a call-back function) for an "infinite" generator based on an Object dictionary as in the following odds-only code written as a JavaScript class:

var SoEIncClass = (function () {
function SoEIncClass() {
this.n = 0;
}
SoEIncClass.prototype.next = function () {
this.n += 2;
if (this.n < 7) { // initialization of sequence to avoid runaway:
if (this.n < 3) { // only even of two:
this.n = 1; // odds from here...
return 2;
}
if (this.n < 5)
return 3;
this.dict = {}; // n must be 5...
this.bps = new SoEIncClass(); // new source of base primes
this.bps.next(); // advance past the even prime of two...
this.p = this.bps.next(); // first odd prime (3 in this case)
this.q = this.p * this.p; // set guard
return 5;
} else { // past initialization:
var s = this.dict[this.n]; // may or may not be defined...
if (!s) { // not defined:
if (this.n < this.q) // haven't reached the guard:
return this.n; // found a prime
else { // n === q => not prime but at guard, so:
var p2 = this.p << 1; // the span odds-only is twice prime
this.dict[this.n + p2] = p2; // add next composite of prime to dict
this.p = this.bps.next();
this.q = this.p * this.p; // get next base prime guard
return this.next(); // not prime so advance...
}
} else { // is a found composite of previous base prime => not prime
delete this.dict[this.n]; // advance to next composite of this prime:
var nxt = this.n + s;
while (this.dict[nxt]) nxt += s; // find unique empty slot in dict
this.dict[nxt] = s; // to put the next composite for this base prime
return this.next(); // not prime so advance...
}
}
};
return SoEIncClass;
})();

The above code can be used to find the nth prime (which would require estimating the required range limit using the previous fixed range code) by using the following code:

var gen = new SoEIncClass(); 
for (var i = 1; i < 1000000; i++, gen.next());
var prime = gen.next();
 
if (typeof print == "undefined")
print = (typeof WScript != "undefined") ? WScript.Echo : alert;
print(prime);

to produce the following output (about five seconds using Google Chrome's V8 JavaScript engine):

15485863

The above code is considerably slower than the fixed range code due to the multiple method calls and the use of an object as a dictionary, which (using a hash table as its basis for most implementations) will have about a constant O(1) amortized time per operation but has quite a high constant overhead to convert the numeric indices to strings which are then hashed to be used as table keys for the look-up operations as compared to doing this more directly in implementations such as the Python dict with Python's built-in hashing functions for every supported type.

This can be implemented as an "infinite" odds-only generator using page segmentation for a considerable speed-up with the alternate JavaScript class code as follows:

var SoEPgClass = (function () {
function SoEPgClass() {
this.bi = -1; // constructor resets the enumeration to start...
}
SoEPgClass.prototype.next = function () {
if (this.bi < 1) {
if (this.bi < 0) {
this.bi++;
this.lowi = 0; // other initialization done here...
this.bpa = [];
return 2;
} else { // bi must be zero:
var nxt = 3 + (this.lowi << 1) + 262144;
this.buf = new Array();
for (var i = 0; i < 4096; i++) // faster initialization:
this.buf.push(0);
if (this.lowi <= 0) { // special culling for first page as no base primes yet:
for (var i = 0, p = 3, sqr = 9; sqr < nxt; i++, p += 2, sqr = p * p)
if ((this.buf[i >> 5] & (1 << (i & 31))) === 0)
for (var j = (sqr - 3) >> 1; j < 131072; j += p)
this.buf[j >> 5] |= 1 << (j & 31);
} else { // after the first page:
if (!this.bpa.length) { // if this is the first page after the zero one:
this.bps = new SoEPgClass(); // initialize separate base primes stream:
this.bps.next(); // advance past the only even prime of two
this.bpa.push(this.bps.next()); // get the next prime (3 in this case)
}
// get enough base primes for the page range...
for (var p = this.bpa[this.bpa.length - 1], sqr = p * p; sqr < nxt;
p = this.bps.next(), this.bpa.push(p), sqr = p * p) ;
for (var i = 0; i < this.bpa.length; i++) {
var p = this.bpa[i];
var s = (p * p - 3) >> 1;
if (s >= this.lowi) // adjust start index based on page lower limit...
s -= this.lowi;
else {
var r = (this.lowi - s) % p;
s = (r != 0) ? p - r : 0;
}
for (var j = s; j < 131072; j += p)
this.buf[j >> 5] |= 1 << (j & 31);
}
}
}
}
while (this.bi < 131072 && this.buf[this.bi >> 5] & (1 << (this.bi & 31)))
this.bi++; // find next marker still with prime status
if (this.bi < 131072) // within buffer: output computed prime
return 3 + ((this.lowi + this.bi++) << 1);
else { // beyond buffer range: advance buffer
this.bi = 0;
this.lowi += 131072;
return this.next(); // and recursively loop
}
};
return SoEPgClass;
})();

The above code is about fifty times faster (about five seconds to calculate 50 million primes to about a billion on the Google Chrome V8 JavaScript engine) than the above dictionary based code.

The speed for both of these "infinite" solutions will also respond to further wheel factorization techniques, especially for the dictionary based version where any added overhead to deal with the factorization wheel will be negligible compared to the dictionary overhead. The dictionary version would likely speed up about a factor of three or a little more with maximum wheel factorization applied; the page segmented version probably won't gain more than a factor of two and perhaps less due to the overheads of array look-up operations.

[edit] jq

Works with: jq version 1.4

Short and sweet ...

# Denoting the input by $n, which is assumed to be a positive integer,
# eratosthenes/0 produces an array of primes less than or equal to $n:
def eratosthenes:
 
# erase(i) sets .[i*j] to false for integral j > 1
def erase(i):
if .[i] then reduce range(2; (1 + length) / i) as $j (.; .[i * $j] = false)
else .
end;
 
(. + 1) as $n
| (($n|sqrt) / 2) as $s
| [null, null, range(2; $n)]
| reduce (2, 1 + (2 * range(1; $s))) as $i (.; erase($i))
| map(select(.));

Examples:

100 | eratosthenes
Output:

[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]

1e7 | eratosthenes | length
Output:

664579

[edit] Liberty BASIC

    'Notice that arrays are globally visible to functions.
'The sieve() function uses the flags() array.
'This is a Sieve benchmark adapted from BYTE 1985
' May, page 286
 
size = 7000
dim flags(7001)
start = time$("ms")
print sieve(size); " primes found."
print "End of iteration. Elapsed time in milliseconds: "; time$("ms")-start
end
 
function sieve(size)
for i = 0 to size
if flags(i) = 0 then
prime = i + i + 3
k = i + prime
while k <= size
flags(k) = 1
k = k + prime
wend
sieve = sieve + 1
end if
next i
end function

[edit]

to sieve :limit
  make "a (array :limit 2)     ; initialized to empty lists
  make "p []
  for [i 2 :limit] [
    if empty? item :i :a [
      queue "p :i
      for [j [:i * :i] :limit :i] [setitem :j :a :i]
    ]
  ]
  output :p
end
print sieve 100   ; 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

[edit] Lua

function erato(n)
if n < 2 then return {} end
local t = {0} -- clears '1'
local sqrtlmt = math.sqrt(n)
for i = 2, n do t[i] = 1 end
for i = 2, sqrtlmt do if t[i] ~= 0 then for j = i*i, n, i do t[j] = 0 end end end
local primes = {}
for i = 2, n do if t[i] ~= 0 then table.insert(primes, i) end end
return primes
end

The following changes the code to odds-only using the same large array-based algorithm:

function erato2(n)
if n < 2 then return {} end
if n < 3 then return {2} end
local t = {}
local lmt = (n - 3) / 2
local sqrtlmt = (math.sqrt(n) - 3) / 2
for i = 0, lmt do t[i] = 1 end
for i = 0, sqrtlmt do if t[i] ~= 0 then
local p = i + i + 3
for j = (p*p - 3) / 2, lmt, p do t[j] = 0 end end end
local primes = {2}
for i = 0, lmt do if t[i] ~= 0 then table.insert(primes, i + i + 3) end end
return primes
end

The following code implements an odds-only "infinite" generator style using a table as a hash table, including postponing adding base primes to the table:

function newEratoInf()
local _cand = 0; local _lstbp = 3; local _lstsqr = 9
local _composites = {}; local _bps = nil
local _self = {}
function _self.next()
if _cand < 9 then if _cand < 1 then _cand = 1; return 2
elseif _cand >= 7 then
--advance aux source base primes to 3...
_bps = newEratoInf()
_bps.next(); _bps.next() end end
_cand = _cand + 2
if _composites[_cand] == nil then -- may be prime
if _cand >= _lstsqr then -- if not the next base prime
local adv = _lstbp + _lstbp -- if next base prime
_composites[_lstbp * _lstbp + adv] = adv -- add cull seq
_lstbp = _bps.next(); _lstsqr = _lstbp * _lstbp -- adv next base prime
return _self.next()
else return _cand end -- is prime
else
local v = _composites[_cand]
_composites[_cand] = nil
local nv = _cand + v
while _composites[nv] ~= nil do nv = nv + v end
_composites[nv] = v
return _self.next() end
end
return _self
end
 
gen = newEratoInf()
count = 0
while gen.next() <= 10000000 do count = count + 1 end -- sieves to 10 million
print(count)
 

which outputs "664579" in about three seconds. As this code uses much less memory for a given range than the previous ones and retains efficiency better with range, it is likely more appropriate for larger sieve ranges.

[edit] Lucid

This example is incorrect. Not a true Sieve of Eratosthenes but rather a Trial Division Sieve Please fix the code and remove this message.
prime
 where
    prime = 2 fby (n whenever isprime(n));
    n = 3 fby n+2;
    isprime(n) = not(divs) asa divs or prime*prime > N
                    where
                      N is current n;
                      divs = N mod prime eq 0;
                    end;
 end

[edit] recursive

This example is incorrect. Not a true Sieve of Eratosthenes but rather a Trial Division Sieve Please fix the code and remove this message.
sieve( N )
   where
    N = 2 fby N + 1;
    sieve( i ) =
      i fby sieve ( i whenever i mod first i ne 0 ) ;
   end

[edit] M4

define(`lim',100)dnl
define(`for',
`ifelse($#,0,
``$0'',
`ifelse(eval($2<=$3),1,
`pushdef(`$1',$2)$5`'popdef(`$1')$0(`$1',eval($2+$4),$3,$4,`$5')')')')dnl
for(`j',2,lim,1,
`ifdef(a[j],
`',
`j for(`k',eval(j*j),lim,j,
`define(a[k],1)')')')
 

Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

[edit] Mathematica

Eratosthenes[n_] := Module[{numbers = Range[n]},
Do[If[numbers[[i]] != 0, Do[numbers[[i j]] = 0, {j, 2, n/i}]], {i,
2, Sqrt[n]}];
Select[numbers, # > 1 &]]
 
Eratosthenes[100]

[edit] Slightly Optimized Version

The below has been improved to not require so many operations per composite number cull for about two thirds the execution time:

Eratosthenes[n_] := Module[{numbers = Range[n]},
Do[If[numbers[[i]] != 0, Do[numbers[[j]] = 0, {j,i i,n,i}]],{i,2,Sqrt[n]}];
Select[numbers, # > 1 &]]
 
Eratosthenes[100]

[edit] Sieving Odds-Only Version

The below has been further improved to only sieve odd numbers for a further reduction in execution time by a factor of over two:

Eratosthenes2[n_] := Module[{numbers = Range[3, n, 2], limit = (n - 1)/2}, 
Do[c = numbers[[i]]; If[c != 0,
Do[numbers[[j]] = 0, {j,(c c - 1)/2,limit,c}]], {i,1,(Sqrt[n] - 1)/2}];
Prepend[Select[numbers, # > 1 &], 2]]
 
Eratosthenes2[100]

[edit] MATLAB

[edit] Somewhat optimized true Sieve of Eratosthenes

 
function P = erato(x) % Sieve of Eratosthenes: returns all primes between 2 and x
 
P = [0 2:x] ; % Create vector with all ints between 2 and x where
% position 1 is hard-coded as 0 since 1 is not a prime.
 
for (n=2:sqrt(x)) % All primes factors lie between 2 and sqrt(x).
if P(n) % If the current value is not 0 (i.e. a prime),
P((2*n):n:x) = 0 ; % then replace all further multiples of it with 0.
end
end % At this point P is a vector with only primes and zeroes.
 
P = P(P ~= 0) ; % Remove all zeroes from P, leaving only the primes.
 
return
The optimization lies in fewer steps in the for loop, use of MATLAB's built-in array operations and no modulo calculation.

Limitation: your machine has to be able to allocate enough memory for an array of length x.

[edit] A more efficient Sieve

A more efficient Sieve avoids creating a large double precision vector P, instead using a logical array (which consumes 1/8 the memory of a double array of the same size) and only converting to double those values corresponding to primes.

 
function P = sieveOfEratosthenes(x)
ISP = [false true(1, x-1)]; % 1 is not prime, but we start off assuming all numbers between 2 and x are
for n = 2:sqrt(x)
if ISP(n)
ISP((2*n):n:x) = false; % Multiples of n that are greater than n are not primes
end
end
% The ISP vector that we have calculated is essentially the output of the ISPRIME function called on 1:x
P = find(ISP); % Convert the ISPRIME output to the values of the primes by finding the locations
% of the TRUE values in S.
 

You can compare the output of this function against the PRIMES function included in MATLAB, which performs a somewhat more memory-efficient Sieve (by not storing even numbers, at the expense of a more complicated indexing expression inside the IF statement.)

[edit] Maxima

sieve(n):=block(
[a:makelist(true,n),i:1,j],
a[1]:false,
do (
i:i+1,
unless a[i] do i:i+1,
if i*i>n then return(sublist_indices(a,identity)),
for j from i*i step i while j<=n do a[j]:false
)
)$

[edit] MAXScript

fn eratosthenes n =
(
    multiples = #()
    print 2
    for i in 3 to n do
    (
        if (findItem multiples i) == 0 then
        (
            print i
            for j in (i * i) to n by i do
            (
                append multiples j
            )
        )
    )
)

eratosthenes 100

[edit] Modula-3

[edit] Regular version

This example is incorrect. Not a true Sieve of Eratosthenes but rather a Trial Division Sieve Please fix the code and remove this message.
MODULE Prime EXPORTS Main;
 
IMPORT IO;
 
CONST LastNum = 1000;
 
VAR a: ARRAY [2..LastNum] OF BOOLEAN;
 
BEGIN
FOR i := FIRST(a) TO LAST(a) DO
a[i] := TRUE;
END;
 
FOR i := FIRST(a) TO LAST(a) DO
IF a[i] THEN
IO.PutInt(i);
IO.Put(" ");
FOR j := FIRST(a) TO LAST(a) DO
IF j MOD i = 0 THEN
a[j] := FALSE;
END;
END;
END;
END;
IO.Put("\n");
 
END Prime.

[edit] Advanced version

This version uses more "advanced" types.

(* From the CM3 examples folder (comments removed). *)
 
MODULE Sieve EXPORTS Main;
 
IMPORT IO;
 
TYPE
Number = [2..1000];
Set = SET OF Number;
 
VAR
prime: Set := Set {FIRST(Number) .. LAST(Number)};
 
BEGIN
FOR i := FIRST(Number) TO LAST(Number) DO
IF i IN prime THEN
IO.PutInt(i);
IO.Put(" ");
 
FOR j := i TO LAST(Number) BY i DO
prime := prime - Set{j};
END;
END;
END;
IO.Put("\n");
END Sieve.

[edit] MUMPS

ERATO1(HI)
 ;performs the Sieve of Erotosethenes up to the number passed in.
 ;This version sets an array containing the primes
SET HI=HI\1
KILL ERATO1 ;Don't make it new - we want it to remain after we quit the function
NEW I,J,P
FOR I=2:1:(HI**.5)\1 FOR J=I*I:I:HI SET P(J)=1
FOR I=2:1:HI S:'$DATA(P(I)) ERATO1(I)=I
KILL I,J,P
QUIT

Example:

USER>SET MAX=100,C=0 DO ERATO1^ROSETTA(MAX) 
USER>WRITE !,"PRIMES BETWEEN 1 AND ",MAX,! FOR  SET I=$ORDER(ERATO1(I)) Q:+I<1  WRITE I,", "

PRIMES BETWEEN 1 AND 100
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,79, 83, 89, 97,

[edit] NetRexx

[edit] Version 1 (slow)

/* NetRexx */
 
options replace format comments java crossref savelog symbols binary
 
parse arg loWatermark hiWatermark .
if loWatermark = '' | loWatermark = '.' then loWatermark = 1
if hiWatermark = '' | hiWatermark = '.' then hiWatermark = 200
 
do
if \loWatermark.datatype('w') | \hiWatermark.datatype('w') then -
signal NumberFormatException('arguments must be whole numbers')
if loWatermark > hiWatermark then -
signal IllegalArgumentException('the start value must be less than the end value')
 
seive = sieveOfEratosthenes(hiWatermark)
primes = getPrimes(seive, loWatermark, hiWatermark).strip
 
say 'List of prime numbers from' loWatermark 'to' hiWatermark 'via a "Sieve of Eratosthenes" algorithm:'
say ' 'primes.changestr(' ', ',')
say ' Count of primes:' primes.words
catch ex = Exception
ex.printStackTrace
end
 
return
 
method sieveOfEratosthenes(hn = long) public static binary returns Rexx
 
sv = Rexx(isTrue)
sv[1] = isFalse
ix = long
jx = long
 
loop ix = 2 while ix * ix <= hn
if sv[ix] then loop jx = ix * ix by ix while jx <= hn
sv[jx] = isFalse
end jx
end ix
 
return sv
 
method getPrimes(seive = Rexx, lo = long, hi = long) private constant binary returns Rexx
 
primes = Rexx('')
loop p_ = lo to hi
if \seive[p_] then iterate p_
primes = primes p_
end p_
 
return primes
 
method isTrue public constant binary returns boolean
return 1 == 1
 
method isFalse public constant binary returns boolean
return \isTrue
 
Output
List of prime numbers from 1 to 200 via a "Sieve of Eratosthenes" algorithm:
  2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199
  Count of primes: 46

[edit] Version 2 (significantly, i.e. 10 times faster)

/* NetRexx ************************************************************
* Essential improvements:Use boolean instead of Rexx for sv
* and remove methods isTrue and isFalse
* 24.07.2012 Walter Pachl courtesy Kermit Kiser
**********************************************************************/

 
options replace format comments java crossref savelog symbols binary
 
parse arg loWatermark hiWatermark .
if loWatermark = '' | loWatermark = '.' then loWatermark = 1
if hiWatermark = '' | hiWatermark = '.' then hiWatermark = 200000
 
startdate=Date Date()
do
if \loWatermark.datatype('w') | \hiWatermark.datatype('w') then -
signal NumberFormatException('arguments must be whole numbers')
if loWatermark > hiWatermark then -
signal IllegalArgumentException(-
'the start value must be less than the end value')
sieve = sieveOfEratosthenes(hiWatermark)
primes = getPrimes(sieve, loWatermark, hiWatermark).strip
if hiWatermark = 200 Then do
say 'List of prime numbers from' loWatermark 'to' hiWatermark
say ' 'primes.changestr(' ', ',')
end
catch ex = Exception
ex.printStackTrace
end
enddate=Date Date()
Numeric Digits 20
say (enddate.getTime-startdate.getTime)/1000 'seconds elapsed'
say ' Count of primes:' primes.words
 
return
 
method sieveOfEratosthenes(hn = int) -
public static binary returns boolean[]
true = boolean 1
false = boolean 0
sv = boolean[hn+1]
sv[1] = false
 
ix = int
jx = int
 
loop ix=2 to hn
sv[ix]=true
end ix
 
loop ix = 2 while ix * ix <= hn
if sv[ix] then loop jx = ix * ix by ix while jx <= hn
sv[jx] = false
end jx
end ix
 
return sv
 
method getPrimes(sieve = boolean[], lo = int, hi = int) -
private constant binary Returns Rexx
p_ = int
primes = Rexx('')
loop p_ = lo to hi
if \sieve[p_] then iterate p_
primes = primes p_
end p_
 
return primes

[edit] Nial

This example is incorrect. It uses rem testing and so is a trial division algorithm, not a sieve of Eratosthenes. Please fix the code and remove this message.
primes is sublist [ each (2 = sum eachright (0 = mod) [pass,count]), pass ] rest count

Using it

|primes 10
=2 3 5 7

[edit] Nim

Based on one of Python solutions:

import math, strutils
 
var is_prime: seq[Bool] = @[]
is_prime.add(False)
is_prime.add(False)
 
iterator iprimes_upto(limit: int): int =
for n in high(is_prime) .. limit+2: is_prime.add(True)
for n in 2 .. limit + 1:
if is_prime[n]:
yield n
for i in countup((n *% n), limit+1, n): # start at ``n`` squared
try:
is_prime[i] = False
except EInvalidIndex: break
 
 
echo("Primes are:")
for x in iprimes_upto(200):
write(stdout, x, " ")
writeln(stdout,"")
Output:
Primes are:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

[edit] Niue

This example is incorrect. It uses rem testing and so is a trial division algorithm, not a sieve of Eratosthenes. Please fix the code and remove this message.
[ dup 2 < ] '<2 ;
[ 1 + 'count ; [ <2 [ , ] when ] count times ] 'fill-stack ;
 
0 'n ; 0 'v ;
 
[ .clr 0 'n ; 0 'v ; ] 'reset ;
[ len 1 - n - at 'v ; ] 'set-base ;
[ n 1 + 'n ; ] 'incr-n ;
[ mod 0 = ] 'is-factor ;
[ dup * ] 'sqr ;
 
[ set-base
v sqr 2 at > not
[ [ dup v = not swap v is-factor and ] remove-if incr-n run ] when ] 'run ;
 
[ fill-stack run ] 'sieve ;
 
( tests )
 
10 sieve .s ( => 2 3 5 7 9 ) reset newline
30 sieve .s ( => 2 3 5 7 11 13 17 19 23 29 )

[edit] Oberon-2

MODULE Primes;
 
IMPORT Out, Math;
 
CONST N = 1000;
 
VAR a: ARRAY N OF BOOLEAN;
i, j, m: INTEGER;
 
BEGIN
(* Set all elements of a to TRUE. *)
FOR i := 1 TO N - 1 DO
a[i] := TRUE;
END;
 
(* Compute square root of N and convert back to INTEGER. *)
m := ENTIER(Math.Sqrt(N));
 
FOR i := 2 TO m DO
IF a[i] THEN
FOR j := 2 TO (N - 1) DIV i DO
a[i*j] := FALSE;
END;
END;
END;
 
(* Print all the elements of a that are TRUE. *)
FOR i := 2 TO N - 1 DO
IF a[i] THEN
Out.Int(i, 4);
END;
END;
Out.Ln;
END Primes.

[edit] OCaml

[edit] Imperative

let sieve n =
let is_prime = Array.create n true in
let limit = truncate(sqrt (float (n - 1))) in
for i = 2 to limit do
if is_prime.(i) then
let j = ref (i*i) in
while !j < n do
is_prime.(!j) <- false;
j := !j + i;
done
done;
is_prime.(0) <- false;
is_prime.(1) <- false;
is_prime
let primes n =
let primes, _ =
let sieve = sieve n in
Array.fold_right
(fun is_prime (xs, i) -> if is_prime then (i::xs, i-1) else (xs, i-1))
sieve
([], Array.length sieve - 1)
in
primes

in the top-level:

# primes 100 ;;
- : int list =
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
 73; 79; 83; 89; 97]

[edit] Functional

(* first define some iterators *)
# let fold_iter f init a b =
let rec aux acc i =
if i > b
then (acc)
else aux (f acc i) (succ i)
in
aux init a ;;
val fold_iter : ('a -> int -> 'a) -> 'a -> int -> int -> 'a = <fun>
 
# let fold_step f init a b step =
let rec aux acc i =
if i > b
then (acc)
else aux (f acc i) (i + step)
in
aux init a ;;
val fold_step : ('a -> int -> 'a) -> 'a -> int -> int -> int -> 'a = <fun>
 
(* remove a given value from a list *)
# let remove li v =
let rec aux acc = function
| hd::tl when hd = v -> (List.rev_append acc tl)
| hd::tl -> aux (hd::acc) tl
| [] -> li
in
aux [] li ;;
val remove : 'a list -> 'a -> 'a list = <fun>
 
(* the main function *)
# let primes n =
let li =
(* create a list [from 2; ... until n] *)
List.rev(fold_iter (fun acc i -> (i::acc)) [] 2 n)
in
let limit = truncate(sqrt(float n)) in
fold_iter (fun li i ->
if List.mem i li (* test if (i) is prime *)
then (fold_step remove li (i*i) n i)
else li)
li 2 (pred limit)
;;
val primes : int -> int list = <fun>
 
# primes 200 ;;
- : int list =
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
73; 79; 83; 89; 97; 101; 103; 107; 109; 113; 127; 131; 137; 139; 149; 151;
157; 163; 167; 173; 179; 181; 191; 193; 197; 199]

[edit] Another functional version

This uses zero to denote struck-out numbers. It is slightly inefficient as it strikes-out multiples above p rather than p2

# let rec strike_nth k n l = match l with
| [] -> []
| h :: t ->
if k = 0 then 0 :: strike_nth (n-1) n t
else h :: strike_nth (k-1) n t;;
val strike_nth : int -> int -> int list -> int list = <fun>
 
# let primes n =
let limit = truncate(sqrt(float n)) in
let rec range a b = if a > b then [] else a :: range (a+1) b in
let rec sieve_primes l = match l with
| [] -> []
| 0 :: t -> sieve_primes t
| h :: t -> if h > limit then List.filter ((<) 0) l else
h :: sieve_primes (strike_nth (h-1) h t) in
sieve_primes (range 2 n) ;;
val primes : int -> int list = <fun>
 
# primes 200;;
- : int list =
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
73; 79; 83; 89; 97; 101; 103; 107; 109; 113; 127; 131; 137; 139; 149; 151;
157; 163; 167; 173; 179; 181; 191; 193; 197; 199]

[edit] Oforth

func: eratosthenes(n)
{
| i j |
ListBuffer newSize(n) dup add(null) seqFrom(2, n) over addAll
2 n sqrt asInteger for: i [
dup at(i) ifNotNull: [ i sq n i step: j [ dup put(j, null) ] ]
]
filter(#notNull)
}
Output:
>eratosthenes(100) println
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

[edit] Oz

Translation of: Haskell
declare
fun {Sieve N}
S = {Array.new 2 N true}
M = {Float.toInt {Sqrt {Int.toFloat N}}}
in
for I in 2..M do
if S.I then
for J in I*I..N;I do
S.J := false
end
end
end
S
end
 
fun {Primes N}
S = {Sieve N}
in
for I in 2..N collect:C do
if S.I then {C I} end
end
end
in
{Show {Primes 30}}

[edit] PARI/GP

Eratosthenes(lim)={
my(v=Vectorsmall(lim\1,unused,1));
forprime(p=2,sqrt(lim),
forstep(i=p^2,lim,p,
v[i]=0
)
);
for(i=1,lim,if(v[i],print1(i", ")))
};

An alternate version:

Sieve(n)=
{
v=vector(n,unused,1);
for(i=2,sqrt(n),
if(v[i],
forstep(j=i^2,n,i,v[j]=0)));
for(i=2,n,if(v[i],print1(i)))
};

[edit] Pascal

Note: Some Pascal implementations put quite low limits on the size of a set (e.g. Turbo Pascal doesn't allow more than 256 members). To compile on such an implementation, reduce the constant PrimeLimit accordingly.

 
program primes(output)
 
const
PrimeLimit = 1000;
 
var
primes: set of 1 .. PrimeLimit;
n, k: integer;
needcomma: boolean;
 
begin
{ calculate the primes }
primes := [2 .. PrimeLimit];
for n := 1 to trunc(sqrt(PrimeLimit)) do
begin
if n in primes
then
begin
k := n*n;
while k < PrimeLimit do
begin
primes := primes - [k];
k := k + n
end
end
end;
 
{ output the primes }
needcomma := false;
for n := 1 to PrimeLimit do
if n in primes
then
begin
if needcomma
then
write(', ');
write(n);
needcomma := true
end
end.
 

[edit] alternative using wheel

Using growing wheel to fill array for sieving for minimal unmark operations. Sieving only with possible-prime factors.

 
program prim(output);
//Sieve of Erathosthenes with fast elimination of multiples of small primes
{$IFNDEF FPC}
{$APPTYPE CONSOLE}
{$ENDIF}
const
PrimeLimit = 100*1000*1000;//1;
type
tLimit = 1..PrimeLimit;
var
//always initialized with 0 => false at startup
primes: array [tLimit] of boolean;
 
function BuildWheel: longInt;
//fill primfield with no multiples of small primes
//returns next sieveprime
//speedup ~1/3
var
//wheelprimes = 2,3,5,7,11... ;
//wheelsize = product [i= 0..wpno-1]wheelprimes[i] > Uint64 i> 13
wheelprimes :array[0..13] of byte;
wheelSize,wpno,
pr,pw,i, k: LongWord;
begin
//the mother of all numbers 1 ;-)
//the first wheel = generator of numbers
//not divisible by the small primes first found primes
pr := 1;
primes[1]:= true;
WheelSize := 1;
 
wpno := 0;
repeat
inc(pr);
//pw = pr projected in wheel of wheelsize
pw := pr;
if pw > wheelsize then
dec(pw,wheelsize);
If Primes[pw] then
begin
// writeln(pr:10,pw:10,wheelsize:16);
k := WheelSize+1;
//turn the wheel (pr-1)-times
for i := 1 to pr-1 do
begin
inc(k,WheelSize);
if k<primeLimit then
move(primes[1],primes[k-WheelSize],WheelSize)
else
begin
move(primes[1],primes[k-WheelSize],PrimeLimit-WheelSize*i);
break;
end;
end;
dec(k);
IF k > primeLimit then
k := primeLimit;
wheelPrimes[wpno] := pr;
primes[pr] := false;
 
inc(wpno);
//the new wheelsize
WheelSize := k;
 
//sieve multiples of the new found prime
i:= pr;
i := i*i;
while i <= k do
begin
primes[i] := false;
inc(i,pr);
end;
end;
until WheelSize >= PrimeLimit;
 
//re-insert wheel-primes
// 1 still stays prime
while wpno > 0 do
begin
dec(wpno);
primes[wheelPrimes[wpno]] := true;
end;
BuildWheel := pr+1;
end;
 
procedure Sieve;
var
sieveprime,
fakt : LongWord;
begin
//primes[1] = true is needed to stop for sieveprime = 2
// at //Search next smaller possible prime
sieveprime := BuildWheel;
//alternative here
//fillchar(primes,SizeOf(Primes),chr(ord(true)));sieveprime := 2;
repeat
if primes[sieveprime] then
begin
//eliminate 'possible prime' multiples of sieveprime
//must go downwards
//2*2 would unmark 4 -> 4*2 = 8 wouldnt be unmarked
fakt := PrimeLimit DIV sieveprime;
IF fakt < sieveprime then
BREAK;
repeat
//Unmark
primes[sieveprime*fakt] := false;
//Search next smaller possible prime
repeat
dec(fakt);
until primes[fakt];
until fakt < sieveprime;
end;
inc(sieveprime);
until false;
//remove 1
primes[1] := false;
end;
 
var
prCnt,
i : LongWord;
Begin
Sieve;
{count the primes }
prCnt := 0;
for i:= 1 to PrimeLimit do
inc(prCnt,Ord(primes[i]));
writeln(prCnt,' primes up to ',PrimeLimit);
end.

output: ( i3 4330 Haswell 3.5 Ghz fpc 2.6.4 -O3 )

5761455 primes up to 100000000

real	0m0.204s
user	0m0.193s
sys	0m0.013s

[edit] Perl

For highest performance and ease, typically a module would be used, such as Math::Prime::Util, Math::Prime::FastSieve, or Math::Prime::XS.

[edit] Classic Sieve

sub sieve {
my $n = shift;
my @composite;
for my $i (2 .. int(sqrt($n))) {
if (!$composite[$i]) {
for (my $j = $i*$i; $j <= $n; $j += $i) {
$composite[$j] = 1;
}
}
}
my @primes;
for my $i (2 .. $n) {
$composite[$i] || push @primes, $i;
}
@primes;
}

[edit] Odds only (faster)

sub sieve2 {
my($n) = @_;
return @{([],[],[2],[2,3],[2,3])[$n]} if $n <= 4;
 
my @composite;
for (my $t = 3; $t*$t <= $n; $t += 2) {
if (!$composite[$t]) {
for (my $s = $t*$t; $s <= $n; $s += $t*2)
{ $composite[$s]++ }
}
}
my @primes = (2);
for (my $t = 3; $t <= $n; $t += 2) {
$composite[$t] || push @primes, $t;
}
@primes;
}

[edit] Odds only, using vectors for lower memory use

sub dj_vector {
my($end) = @_;
return @{([],[],[2],[2,3],[2,3])[$end]} if $end <= 4;
$end-- if ($end & 1) == 0; # Ensure end is odd
 
my ($sieve, $n, $limit, $s_end) = ( '', 3, int(sqrt($end)), $end >> 1 );
while ( $n <= $limit ) {
for (my $s = ($n*$n) >> 1; $s <= $s_end; $s += $n) {
vec($sieve, $s, 1) = 1;
}
do { $n += 2 } while vec($sieve, $n >> 1, 1) != 0;
}
my @primes = (2);
do { push @primes, 2*$_+1 if !vec($sieve,$_,1) } for (1..int(($end-1)/2));
@primes;
}

[edit] Odds only, using strings for best performance

Compared to array versions, about 2x faster (with 5.16.0 or later) and lower memory. Much faster than the experimental versions below. It's possible a mod-6 or mod-30 wheel could give more improvement, though possibly with obfuscation. The best next step for performance and functionality would be segmenting.

sub string_sieve {
my ($n, $i, $s, $d, @primes) = (shift, 7);
 
local $_ = '110010101110101110101110111110' .
'101111101110101110101110111110' x ($n/30);
 
until (($s = $i*$i) > $n) {
$d = $i<<1;
do { substr($_, $s, 1, '1') } until ($s += $d) > $n;
1 while substr($_, $i += 2, 1);
}
$_ = substr($_, 1, $n);
# For just the count: return ($_ =~ tr/0//);
push @primes, pos while m/0/g;
@primes;
}

This older version uses half the memory, but at the expense of a bit of speed and code complexity:

sub dj_string {
my($end) = @_;
return @{([],[],[2],[2,3],[2,3])[$end]} if $end <= 4;
$end-- if ($end & 1) == 0;
my $s_end = $end >> 1;
 
my $whole = int( ($end>>1) / 15); # prefill with 3 and 5 marked
my $sieve = '100010010010110' . '011010010010110' x $whole;
substr($sieve, ($end>>1)+1) = '';
my ($n, $limit, $s) = ( 7, int(sqrt($end)), 0 );
while ( $n <= $limit ) {
for ($s = ($n*$n) >> 1; $s <= $s_end; $s += $n) {
substr($sieve, $s, 1) = '1';
}
do { $n += 2 } while substr($sieve, $n>>1, 1);
}
# If you just want the count, it's very fast:
# my $count = 1 + $sieve =~ tr/0//;
my @primes = (2);
push @primes, 2*pos($sieve)-1 while $sieve =~ m/0/g;
@primes;
}

[edit] Experimental

These are examples of golfing or unusual styles.

Golfing a bit, at the expense of speed:

sub sieve{ my (@s, $i);
grep { not $s[ $i = $_ ] and do
{ $s[ $i += $_ ]++ while $i <= $_[0]; 1 }
} 2..$_[0]
}
 
print join ", " => sieve 100;

Or with bit strings (much slower than the vector version above):

sub sieve{ my ($s, $i);
grep { not vec $s, $i = $_, 1 and do
{ (vec $s, $i += $_, 1) = 1 while $i <= $_[0]; 1 }
} 2..$_[0]
}
 
print join ", " => sieve 100;

A short recursive version:

sub erat {
my $p = shift;
return $p, $p**2 > $_[$#_] ? @_ : erat(grep $_%$p, @_)
}
 
print join ', ' => erat 2..100000;
Regexp (purely an example -- the regex engine limits it to only 32769):
sub sieve {
my ($s, $p) = "." . ("x" x shift);
 
1 while ++$p
and $s =~ /^(.{$p,}?)x/g
and $p = length($1)
and $s =~ s/(.{$p})./${1}./g
and substr($s, $p, 1) = "x";
$s
}
 
print sieve(1000);

[edit] Extensible sieves

Here are two incremental versions, which allows one to create a tied array of primes:

use strict;
use warnings;
package Tie::SieveOfEratosthenes;
 
sub TIEARRAY {
my $class = shift;
bless \$class, $class;
}
 
# If set to true, produces copious output. Observing this output
# is an excellent way to gain insight into how the algorithm works.
use constant DEBUG => 0;
 
# If set to true, causes the code to skip over even numbers,
# improving runtime. It does not alter the output content, only the speed.
use constant WHEEL2 => 0;
 
BEGIN {
 
# This is loosely based on the Python implementation of this task,
# specifically the "Infinite generator with a faster algorithm"
 
my @primes = (2, 3);
my $ps = WHEEL2 ? 1 : 0;
my $p = $primes[$ps];
my $q = $p*$p;
my $incr = WHEEL2 ? 2 : 1;
my $candidate = $primes[-1] + $incr;
my %sieve;
 
print "Initial: p = $p, q = $q, candidate = $candidate\n" if DEBUG;
 
sub FETCH {
my $n = pop;
return if $n < 0;
return $primes[$n] if $n <= $#primes;
OUTER: while( 1 ) {
 
# each key in %sieve is a composite number between
# p and p-squared. Each value in %sieve is $incr x the prime
# which acted as a 'seed' for that key. We use the value
# to step through multiples of the seed-prime, until we find
# an empty slot in %sieve.
while( my $s = delete $sieve{$candidate} ) {
print "$candidate a multiple of ".($s/$incr).";\t\t" if DEBUG;
my $composite = $candidate + $s;
$composite += $s while exists $sieve{$composite};
print "The next stored multiple of ".($s/$incr)." is $composite\n" if DEBUG;
$sieve{$composite} = $s;
$candidate += $incr;
}
 
print "Candidate $candidate is not in sieve\n" if DEBUG;
 
while( $candidate < $q ) {
print "$candidate is prime\n" if DEBUG;
push @primes, $candidate;
$candidate += $incr;
next OUTER if exists $sieve{$candidate};
}
 
die "Candidate = $candidate, p = $p, q = $q" if $candidate > $q;
print "Candidate $candidate is equal to $p squared;\t" if DEBUG;
 
# Thus, it is now time to add p to the sieve,
my $step = $incr * $p;
my $composite = $q + $step;
$composite += $step while exists $sieve{$composite};
print "The next multiple of $p is $composite\n" if DEBUG;
$sieve{$composite} = $step;
 
# and fetch out a new value for p from our primes array.
$p = $primes[++$ps];
$q = $p * $p;
 
# And since $candidate was equal to some prime squared,
# it's obviously composite, and we need to increment it.
$candidate += $incr;
print "p is $p, q is $q, candidate is $candidate\n" if DEBUG;
} continue {
return $primes[$n] if $n <= $#primes;
}
}
 
}
 
if( !caller ) {
tie my (@prime_list), 'Tie::SieveOfEratosthenes';
my $limit = $ARGV[0] || 100;
my $line = "";
for( my $count = 0; $prime_list[$count] < $limit; ++$count ) {
$line .= $prime_list[$count]. ", ";
next if length($line) <= 70;
if( $line =~ tr/,// > 1 ) {
$line =~ s/^(.*,) (.*, )/$2/;
print $1, "\n";
} else {
print $line, "\n";
$line = "";
}
}
$line =~ s/, \z//;
print $line, "\n" if $line;
}
 
1;

This one is based on the vector sieve shown earlier, but adds to a list as needed, just sieving in the segment. Slightly faster and half the memory vs. the previous incremental sieve. It uses the same API -- arguably we should be offset by one so $primes[$n] returns the $n'th prime.

use strict;
use warnings;
package Tie::SieveOfEratosthenes;
 
sub TIEARRAY {
my $class = shift;
my @primes = (2,3,5,7);
return bless \@primes, $class;
}
 
sub prextend { # Extend the given list of primes using a segment sieve
my($primes, $to) = @_;
$to-- unless $to & 1; # Ensure end is odd
return if $to < $primes->[-1];
my $sqrtn = int(sqrt($to)+0.001);
prextend($primes, $sqrtn) if $primes->[-1] < $sqrtn;
my($segment, $startp) = ('', $primes->[-1]+1);
my($s_beg, $s_len) = ($startp >> 1, ($to>>1) - ($startp>>1));
for my $p (@$primes) {
last if $p > $sqrtn;
if ($p >= 3) {
my $p2 = $p*$p;
if ($p2 < $startp) { # Bump up to next odd multiple of p >= startp
my $f = 1+int(($startp-1)/$p);
$p2 = $p * ($f | 1);
}
for (my $s = ($p2>>1)-$s_beg; $s <= $s_len; $s += $p) {
vec($segment, $s, 1) = 1; # Mark composites in segment
}
}
}
# Now add all the primes found in the segment to the list
do { push @$primes, 1+2*($_+$s_beg) if !vec($segment,$_,1) } for 0 .. $s_len;
}
 
sub FETCHSIZE { 0x7FFF_FFFF } # Allows foreach to work
sub FETCH {
my($primes, $n) = @_;
return if $n < 0;
# Keep expanding the list as necessary, 5% larger each time.
prextend($primes, 1000+int(1.05*$primes->[-1])) while $n > $#$primes;
return $primes->[$n];
}
 
if( !caller ) {
tie my @prime_list, 'Tie::SieveOfEratosthenes';
my $limit = $ARGV[0] || 100;
print $prime_list[0];
my $i = 1;
while ($prime_list[$i] < $limit) { print " ", $prime_list[$i++]; }
print "\n";
}
 
1;

[edit] Perl 6

sub sieve( Int $limit ) {
my @is-prime = False, False, True xx $limit - 1;
 
gather for @is-prime.kv -> $number, $is-prime {
if $is-prime {
take $number;
@is-prime[$_] = False if $_ %% $number for $number**2 .. $limit;
}
}
}
 
(sieve 100).join(",").say;

A recursive version:

multi erat(Int $N) { erat 2 .. $N }
multi erat(@a where @a[0] > sqrt @a[*-1]) { @a }
multi erat(@a) { @a[0], erat(@a.grep: * % @a[0]) }
 
say erat 100;

Of course, upper limits are for wusses. Here's a version using infinite streams, that just keeps going until you ^C it (works under Niecza):

role Filter[Int $factor] {
method next { repeat until $.value % $factor { callsame } }
}
 
class Stream {
has Int $.value is rw = 1;
 
method next { ++$.value }
method filter { self but Filter[$.value] }
}
 
.next, say .value for Stream.new, *.filter ... *;

[edit] PHP

 
function iprimes_upto($limit)
{
for ($i = 2; $i < $limit; $i++)
{
$primes[$i] = true;
}
 
for ($n = 2; $n < $limit; $n++)
{
if ($primes[$n])
{
for ($i = $n*$n; $i < $limit; $i += $n)
{
$primes[$i] = false;
}
}
}
 
return $primes;
}
 

[edit] PicoLisp

(de sieve (N)
(let Sieve (range 1 N)
(set Sieve)
(for I (cdr Sieve)
(when I
(for (S (nth Sieve (* I I)) S (nth (cdr S) I))
(set S) ) ) )
(filter bool Sieve) ) )

Output:

: (sieve 100)
-> (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97)

[edit] PL/I

eratos: proc options (main) reorder;
 
dcl i fixed bin (31);
dcl j fixed bin (31);
dcl n fixed bin (31);
dcl sn fixed bin (31);
 
dcl hbound builtin;
dcl sqrt builtin;
 
dcl sysin file;
dcl sysprint file;
 
get list (n);
sn = sqrt(n);
 
begin;
dcl primes(n) bit (1) aligned init ((*)((1)'1'b));
 
i = 2;
 
do while(i <= sn);
do j = i ** 2 by i to hbound(primes, 1);
/* Adding a test would just slow down processing! */
primes(j) = '0'b;
end;
 
do i = i + 1 to sn until(primes(i));
end;
end;
 
do i = 2 to hbound(primes, 1);
if primes(i) then
put data(i);
end;
end;
end eratos;

[edit] Pop11

define eratostenes(n);
lvars bits = inits(n), i, j;
for i from 2 to n do
   if bits(i) = 0 then
      printf('' >< i, '%s\n');
      for j from 2*i by i to n do
         1 -> bits(j);
      endfor;
   endif;
endfor;
enddefine;


[edit] PowerShell

[edit] Basic procedure

It outputs immediately so that the number can be used by the pipeline.

function Sieve ( [int] $num )
{
$isprime = @{}
2..$num | Where-Object {
$isprime[$_] -eq $null } | ForEach-Object {
$_
$isprime[$_] = $true
$i=$_*$_
for ( ; $i -le $num; $i += $_ )
{ $isprime[$i] = $false }
}
}

[edit] Processing

Calculate the primes up to 1000000 with Processing, including a visualisation of the process. As an additional visual effect, the layout of the pixel could be changed from the line-by-line layout to a spiral-like layout starting in the middle of the screen.

int maxx,maxy;
int max;
boolean[] sieve;
 
void plot(int pos, boolean active) {
set(pos%maxx,pos/maxx, active?#000000:#ffffff);
}
 
void setup() {
size(1000, 1000, P2D);
frameRate(2);
maxx=width;
maxy=height;
max=width*height;
sieve=new boolean[max+1];
 
sieve[1]=false;
plot(0,false);
plot(1,false);
for(int i=2;i<=max;i++) {
sieve[i]=true;
plot(i,true);
}
}
 
int i=2;
 
void draw() {
if(!sieve[i]) {
while(i*i<max && !sieve[i]) {
i++;
}
}
if(sieve[i]) {
print(i+" ");
for(int j=i*i;j<=max;j+=i) {
if(sieve[j]) {
sieve[j]=false;
plot(j,false);
}
}
}
if(i*i<max) {
i++;
} else {
noLoop();
println("finished");
}
}

[edit] Prolog

[edit] Using facts to record composite numbers

The first two solutions use Prolog "facts" to record the composite (i.e. already-visited) numbers.

[edit] Elementary approach: multiplication-free, division-free, mod-free, and cut-free

The basic Eratosthenes sieve depends on nothing more complex than counting. In celebration of this simplicity, the first approach to the problem taken here is free of multiplication and division, as well as Prolog's non-logical "cut".

It defines the predicate between/4 to avoid division, and composite/1 to record integers that are found to be composite.

% %sieve( +N, -Primes ) is true if Primes is the list of consecutive primes
% that are less than or equal to N
sieve( N, [2|Rest]) :-
retractall( composite(_) ),
sieve( N, 2, Rest ) -> true. % only one solution
 
% sieve P, find the next non-prime, and then recurse:
sieve( N, P, [I|Rest] ) :-
sieve_once(P, N),
(P = 2 -> P2 is P+1; P2 is P+2),
between(P2, N, I),
(composite(I) -> fail; sieve( N, I, Rest )).
 
% It is OK if there are no more primes less than or equal to N:
sieve( N, P, [] ).
 
sieve_once(P, N) :-
forall( between(P, N, P, IP),
(composite(IP) -> true ; assertz( composite(IP) )) ).
 
 
% To avoid division, we use the iterator
% between(+Min, +Max, +By, -I)
% where we assume that By > 0
% This is like "for(I=Min; I <= Max; I+=By)" in C.
between(Min, Max, By, I) :-
Min =< Max,
A is Min + By,
(I = Min; between(A, Max, By, I) ).
 
 
% Some Prolog implementations require the dynamic predicates be
% declared:
 
:- dynamic( composite/1 ).
 

The above has been tested with SWI-Prolog and gprolog.

% SWI-Prolog:
 
?- time( (sieve(100000,P), length(P,N), writeln(N), last(P, LP), writeln(LP) )).
% 1,323,159 inferences, 0.862 CPU in 0.921 seconds (94% CPU, 1534724 Lips)
P = [2, 3, 5, 7, 11, 13, 17, 19, 23|...],
N = 9592,
LP = 99991.
 

[edit] Optimized approach

Works with SWI-Prolog.

sieve(N, [2|PS]) :-       % PS is list of odd primes up to N
retractall(mult(_)),
sieve_O(3,N,PS).
 
sieve_O(I,N,PS) :- % sieve odds from I up to N to get PS
I =< N, !, I1 is I+2,
( mult(I) -> sieve_O(I1,N,PS)
; ( I =< N / I ->
ISq is I*I, DI is 2*I, add_mults(DI,ISq,N)
; true
),
PS = [I|T],
sieve_O(I1,N,T)
).
sieve_O(I,N,[]) :- I > N.
 
add_mults(DI,I,N) :-
I =< N, !,
( mult(I) -> true ; assert(mult(I)) ),
I1 is I+DI,
add_mults(DI,I1,N).
add_mults(_,I,N) :- I > N.
 
main(N) :- current_prolog_flag(verbose,F),
set_prolog_flag(verbose,normal),
time( sieve( N,P)), length(P,Len), last(P, LP), writeln([Len,LP]),
set_prolog_flag(verbose,F).
 
:- dynamic( mult/1 ).
:- main(100000), main(1000000).

Running it produces

%% stdout copy
[9592, 99991]
[78498, 999983]
 
%% stderr copy
% 293,176 inferences, 0.14 CPU in 0.14 seconds (101% CPU, 2094114 Lips)
% 3,122,303 inferences, 1.63 CPU in 1.67 seconds (97% CPU, 1915523 Lips)

which indicates ~ N1.1 empirical orders of growth, which is consistent with the O(N log log N) theoretical runtime complexity.

[edit] Using a priority queue

Uses a ariority queue, from the paper "The Genuine Sieve of Eratosthenes" by Melissa O'Neill. Works with YAP (Yet Another Prolog)

?- use_module(library(heaps)).
 
prime(2).
prime(N) :- prime_heap(N, _).
 
prime_heap(3, H) :- list_to_heap([9-6], H).
prime_heap(N, H) :-
prime_heap(M, H0), N0 is M + 2,
next_prime(N0, H0, N, H).
 
next_prime(N0, H0, N, H) :-
\+ min_of_heap(H0, N0, _),
N = N0, Composite is N*N, Skip is N+N,
add_to_heap(H0, Composite, Skip, H).
next_prime(N0, H0, N, H) :-
min_of_heap(H0, N0, _),
adjust_heap(H0, N0, H1), N1 is N0 + 2,
next_prime(N1, H1, N, H).
 
adjust_heap(H0, N, H) :-
min_of_heap(H0, N, _),
get_from_heap(H0, N, Skip, H1),
Composite is N + Skip, add_to_heap(H1, Composite, Skip, H2),
adjust_heap(H2, N, H).
adjust_heap(H, N, H) :-
\+ min_of_heap(H, N, _).

[edit] PureBasic

[edit] Basic procedure

For n=2 To Sqr(lim)
If Nums(n)=0
m=n*n
While m<=lim
Nums(m)=1
m+n
Wend
EndIf
Next n

[edit] Working example

Dim Nums.i(0)
Define l, n, m, lim
 
If OpenConsole()
 
; Ask for the limit to search, get that input and allocate a Array
Print("Enter limit for this search: ")
lim=Val(Input())
ReDim Nums(lim)
 
; Use a basic Sieve of Eratosthenes
For n=2 To Sqr(lim)
If Nums(n)=#False
m=n*n
While m<=lim
Nums(m)=#True
m+n
Wend
EndIf
Next n
 
;Present the result to our user
PrintN(#CRLF$+"The Prims up to "+Str(lim)+" are;")
m=0: l=Log10(lim)+1
For n=2 To lim
If Nums(n)=#False
Print(RSet(Str(n),l)+" ")
m+1
If m>72/(l+1)
m=0: PrintN("")
EndIf
EndIf
Next
 
Print(#CRLF$+#CRLF$+"Press ENTER to exit"): Input()
CloseConsole()
EndIf

Output may look like;

Enter limit for this search: 750

The Prims up to 750 are;
   2    3    5    7   11   13   17   19   23   29   31   37   41   43   47
  53   59   61   67   71   73   79   83   89   97  101  103  107  109  113
 127  131  137  139  149  151  157  163  167  173  179  181  191  193  197
 199  211  223  227  229  233  239  241  251  257  263  269  271  277  281
 283  293  307  311  313  317  331  337  347  349  353  359  367  373  379
 383  389  397  401  409  419  421  431  433  439  443  449  457  461  463
 467  479  487  491  499  503  509  521  523  541  547  557  563  569  571
 577  587  593  599  601  607  613  617  619  631  641  643  647  653  659
 661  673  677  683  691  701  709  719  727  733  739  743

Press ENTER to exit

[edit] Python

Note that the examples use range instead of xrange for Python 3 and Python 2 compatability, but when using Python 2 xrange is the nearest equivalent to Python 3's implementation of range and should be substituted for ranges with a very large number of items.

[edit] Using set lookup

The version below uses a set to store the multiples. set objects are much faster (usually O(log n)) than lists (O(n)) for checking if a given object is a member. Using the set.update method avoids explicit iteration in the interpreter, giving a further speed improvement.

def eratosthenes2(n):
multiples = set()
for i in range(2, n+1):
if i not in multiples:
yield i
multiples.update(range(i*i, n+1, i))
 
print(list(eratosthenes2(100)))

[edit] Using array lookup

The version below uses array lookup to test for primality. The function primes_upto() is a straightforward implementation of Sieve of Eratosthenesalgorithm. It returns prime numbers less than or equal to limit.

def primes_upto(limit):
is_prime = [False] * 2 + [True] * (limit - 1)
for n in range(int(limit**0.5 + 1.5)): # stop at ``sqrt(limit)``
if is_prime[n]:
for i in range(n*n, limit+1, n):
is_prime[i] = False
return [i for i, prime in enumerate(is_prime) if prime]

[edit] Using generator

The following code may be slightly slower than using the array/list as above, but uses no memory for output:

def iprimes_upto(limit):
is_prime = [False] * 2 + [True] * (limit - 1)
for n in xrange(int(limit**0.5 + 1.5)): # stop at ``sqrt(limit)``
if is_prime[n]:
for i in range(n * n, limit + 1, n): # start at ``n`` squared
is_prime[i] = False
for i in xrange(limit + 1):
if is_prime[i]: yield i
Example:
>>> list(iprimes_upto(15))
[2, 3, 5, 7, 11, 13]

[edit] Odds-only version of the array sieve above

The following code is faster than the above array version using only odd composite operations (for a factor of over two) and because it has been optimized to use slice operations for composite number culling to avoid extra work by the interpreter:

def primes2(limit):
if limit < 2: return []
if limit < 3: return [2]
lmtbf = (limit - 3) // 2
buf = [True] * (lmtbf + 1)
for i in range((int(limit ** 0.5) - 3) // 2 + 1):
if buf[i]:
p = i + i + 3
s = p * (i + 1) + i
buf[s::p] = [False] * ((lmtbf - s) // p + 1)
return [2] + [i + i + 3 for i, v in enumerate(buf) if v]

Note that "range" needs to be changed to "xrange" for maximum speed with Python 2.

[edit] Odds-only version of the generator version above

The following code is faster than the above generator version using only odd composite operations (for a factor of over two) and because it has been optimized to use slice operations for composite number culling to avoid extra work by the interpreter:

def iprimes2(limit):
yield 2
if limit < 3: return
lmtbf = (limit - 3) // 2
buf = [True] * (lmtbf + 1)
for i in range((int(limit ** 0.5) - 3) // 2 + 1):
if buf[i]:
p = i + i + 3
s = p * (i + 1) + i
buf[s::p] = [False] * ((lmtbf - s) // p + 1)
for i in range(lmtbf + 1):
if buf[i]: yield (i + i + 3)

Note that this version may actually run slightly faster than the equivalent array version with the advantage that the output doesn't require any memory.

Also note that "range" needs to be changed to "xrange" for maximum speed with Python 2.

[edit] Factorization wheel235 version of the generator version

This uses a 235 factorial wheel for further reductions in operations; the same techniques can be applied to the array version as well; it runs slightly faster and uses slightly less memory as compared to the odds-only algorithms:

def primes235(limit):
yield 2; yield 3; yield 5
if limit < 7: return
modPrms = [7,11,13,17,19,23,29,31]
gaps = [4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6] # 2 loops for overflow
ndxs = [0,0,0,0,1,1,2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,7,7,7,7,7,7]
lmtbf = (limit + 23) // 30 * 8 - 1 # integral number of wheels rounded up
lmtsqrt = (int(limit ** 0.5) - 7)
lmtsqrt = lmtsqrt // 30 * 8 + ndxs[lmtsqrt % 30] # round down on the wheel
buf = [True] * (lmtbf + 1)
for i in range(lmtsqrt + 1):
if buf[i]:
ci = i & 7; p = 30 * (i >> 3) + modPrms[ci]
s = p * p - 7; p8 = p << 3
for j in range(8):
c = s // 30 * 8 + ndxs[s % 30]
buf[c::p8] = [False] * ((lmtbf - c) // p8 + 1)
s += p * gaps[ci]; ci += 1
for i in range(lmtbf - 6 + (ndxs[(limit - 7) % 30])): # adjust for extras
if buf[i]: yield (30 * (i >> 3) + modPrms[i & 7])

Note: Much of the time (almost two thirds for this last case for Python 2.7.6) for any of these array/list or generator algorithms is used in the computation and enumeration of the final output in the last line(s), so any slight changes to those lines can greatly affect execution time. For Python 3 this enumeration is about twice as slow as Python 2 (Python 3.3 slow and 3.4 slower) for an even bigger percentage of time spent just outputting the results. This slow enumeration means that there is little advantage to versions that use even further wheel factorization, as the composite number culling is a small part of the time to enumerate the results.

If just the count of the number of primes over a range is desired, then converting the functions to prime counting functions by changing the final enumeration lines to "return buf.count(True)" will save a lot of time.

Note that "range" needs to be changed to "xrange" for maximum speed with Python 2 where Python 2's "xrange" is a better choice for very large sieve ranges.
Timings were done primarily in Python 2 although source is Python 2/3 compatible (shows range and not xrange).

[edit] Using numpy

Library: numpy

Below code adapted from literateprograms.org using numpy

import numpy
def primes_upto2(limit):
is_prime = numpy.ones(limit + 1, dtype=numpy.bool)
for n in xrange(2, int(limit**0.5 + 1.5)):
if is_prime[n]:
is_prime[n*n::n] = 0
return numpy.nonzero(is_prime)[0][2:]

Performance note: there is no point to add wheels here, due to execution of p[n*n::n] = 0 and nonzero() takes us almost all time.

Also see Prime numbers and Numpy – Python.

[edit] Using wheels with numpy

Version with wheel based optimization:

from numpy import array, bool_, multiply, nonzero, ones, put, resize
#
def makepattern(smallprimes):
pattern = ones(multiply.reduce(smallprimes), dtype=bool_)
pattern[0] = 0
for p in smallprimes:
pattern[p::p] = 0
return pattern
#
def primes_upto3(limit, smallprimes=(2,3,5,7,11)):
sp = array(smallprimes)
if limit <= sp.max(): return sp[sp <= limit]
#
isprime = resize(makepattern(sp), limit + 1)
isprime[:2] = 0; put(isprime, sp, 1)
#
for n in range(sp.max() + 2, int(limit**0.5 + 1.5), 2):
if isprime[n]:
isprime[n*n::n] = 0
return nonzero(isprime)[0]

Examples:

>>> primes_upto3(10**6, smallprimes=(2,3)) # Wall time: 0.17
array([ 2, 3, 5, ..., 999961, 999979, 999983])
>>> primes_upto3(10**7, smallprimes=(2,3)) # Wall time: '''2.13'''
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])
>>> primes_upto3(15)
array([ 2, 3, 5, 7, 11, 13])
>>> primes_upto3(10**7, smallprimes=primes_upto3(15)) # Wall time: '''1.31'''
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])
>>> primes_upto2(10**7) # Wall time: '''1.39''' <-- version ''without'' wheels
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])
>>> primes_upto3(10**7) # Wall time: '''1.30'''
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])

The above-mentioned examples demonstrate that the given wheel based optimization does not show significant performance gain.

[edit] Infinite generator

A generator that will generate primes indefinitely (perhaps until it runs out of memory). Used as a library here.

Works with: Python version 2.6+, 3.x
import heapq
 
# generates all prime numbers
def sieve():
# priority queue of the sequences of non-primes
# the priority queue allows us to get the "next" non-prime quickly
nonprimes = []
 
i = 2
while True:
if nonprimes and i == nonprimes[0][0]: # non-prime
while nonprimes[0][0] == i:
# for each sequence that generates this number,
# have it go to the next number (simply add the prime)
# and re-position it in the priority queue
x = nonprimes[0]
x[0] += x[1]
heapq.heapreplace(nonprimes, x)
 
else: # prime
# insert a 2-element list into the priority queue:
# [current multiple, prime]
# the first element allows sorting by value of current multiple
# we start with i^2
heapq.heappush(nonprimes, [i*i, i])
yield i
 
i += 1

Example:

>>> foo = sieve()
>>> for i in range(8):
...     print(next(foo))
... 
2
3
5
7
11
13
17
19

[edit] Infinite generator with a faster algorithm

The adding of each discovered prime's incremental step info to the mapping should be postponed until the prime's square is seen amongst the candidate numbers, as it is useless before that point. This drastically reduces the space complexity from O(n) to O(sqrt(n/log(n))), in n primes produced, and also lowers the run time complexity quite low (this test entry in Python 2.7 and this test entry in Python 3.x shows about ~ n1.08 empirical order of growth which is very close to the theoretical value of O(n log(n) log(log(n))), in n primes produced):

Works with: Python version 2.6+, 3.x
def primes():
yield 2; yield 3; yield 5; yield 7;
bps = (p for p in primes()) # separate supply of "base" primes (b.p.)
p = next(bps) and next(bps) # discard 2, then get 3
q = p * p # 9 - square of next base prime to keep track of,
sieve = {} # in the sieve dict
n = 9 # n is the next candidate number
while True:
if n not in sieve: # n is not a multiple of any of base primes,
if n < q: # below next base prime's square, so
yield n # n is prime
else:
p2 = p + p # n == p * p: for prime p, add p * p + 2 * p
sieve[q + p2] = p2 # to the dict, with 2 * p as the increment step
p = next(bps); q = p * p # pull next base prime, and get its square
else:
s = sieve.pop(n); nxt = n + s # n's a multiple of some b.p., find next multiple
while nxt in sieve: nxt += s # ensure each entry is unique
sieve[nxt] = s # nxt is next non-marked multiple of this prime
n += 2 # work on odds only
 
import itertools
def primes_up_to(limit):
return list(itertools.takewhile(lambda p: p <= limit, primes()))

[edit] Fast infinite generator using a wheel

Although theoretically over three times faster than odds-only, the following code using a 2/3/5/7 wheel is only about 1.5 times faster than the above odds-only code due to the extra overheads in code complexity. The test link for Python 2.7 and test link for Python 3.x show about the same empirical order of growth as the odds-only implementation above once the range grows enough so the dict operations become amortized to a constant factor.

Works with: Python version 2.6+, 3.x
def primes():
for p in [2,3,5,7]: yield p # base wheel primes
gaps1 = [ 2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,8 ]
gaps = gaps1 + [ 6,4,6,2,4,6,2,6,6,4,2,4,6,2,6,4,2,4,2,10,2,10 ] # wheel2357
def wheel_prime_pairs():
yield (11,0); bps = wheel_prime_pairs() # additional primes supply
p, pi = next(bps); q = p * p # adv to get 11 sqr'd is 121 as next square to put
sieve = {}; n = 13; ni = 1 # into sieve dict; init cndidate, wheel ndx
while True:
if n not in sieve: # is not a multiple of previously recorded primes
if n < q: yield (n, ni) # n is prime with wheel modulo index
else:
npi = pi + 1 # advance wheel index
if npi > 47: npi = 0
sieve[q + p * gaps[pi]] = (p, npi) # n == p * p: put next cull position on wheel
p, pi = next(bps); q = p * p # advance next prime and prime square to put
else:
s, si = sieve.pop(n)
nxt = n + s * gaps[si] # move current cull position up the wheel
si = si + 1 # advance wheel index
if si > 47: si = 0
while nxt in sieve: # ensure each entry is unique by wheel
nxt += s * gaps[si]
si = si + 1 # advance wheel index
if si > 47: si = 0
sieve[nxt] = (s, si) # next non-marked multiple of a prime
nni = ni + 1 # advance wheel index
if nni > 47: nni = 0
n += gaps[ni]; ni = nni # advance on the wheel
for p, pi in wheel_prime_pairs(): yield p # strip out indexes

Further gains of about 1.5 times in speed can be made using the same code by only changing the tables and a few constants for a further constant factor gain of about 1.5 times in speed by using a 2/3/5/7/11/13/17 wheel (with the gaps list 92160 elements long) computed for a slight constant overhead time as per the test link for Python 2.7 and test link for Python 3.x. Further wheel factorization will not really be worth it as the gains will be small (if any and not losses) and the gaps table huge - it is already too big for efficient use by 32-bit Python 3 and the wheel should likely be stopped at 13:

def primes():
whlPrms = [2,3,5,7,11,13,17] # base wheel primes
for p in whlPrms: yield p
def makeGaps():
buf = [True] * (3 * 5 * 7 * 11 * 13 * 17 + 1) # all odds plus extra for o/f
for p in whlPrms:
if p < 3:
continue # no need to handle evens
strt = (p * p - 19) >> 1 # start position (divided by 2 using shift)
while strt < 0: strt += p
buf[strt::p] = [False] * ((len(buf) - strt - 1) // p + 1) # cull for p
whlPsns = [i + i for i,v in enumerate(buf) if v]
return [whlPsns[i + 1] - whlPsns[i] for i in range(len(whlPsns) - 1)]
gaps = makeGaps() # big wheel gaps
def wheel_prime_pairs():
yield (19,0); bps = wheel_prime_pairs() # additional primes supply
p, pi = next(bps); q = p * p # adv to get 11 sqr'd is 121 as next square to put
sieve = {}; n = 23; ni = 1 # into sieve dict; init cndidate, wheel ndx
while True:
if n not in sieve: # is not a multiple of previously recorded primes
if n < q: yield (n, ni) # n is prime with wheel modulo index
else:
npi = pi + 1 # advance wheel index
if npi > 92159: npi = 0
sieve[q + p * gaps[pi]] = (p, npi) # n == p * p: put next cull position on wheel
p, pi = next(bps); q = p * p # advance next prime and prime square to put
else:
s, si = sieve.pop(n)
nxt = n + s * gaps[si] # move current cull position up the wheel
si = si + 1 # advance wheel index
if si > 92159: si = 0
while nxt in sieve: # ensure each entry is unique by wheel
nxt += s * gaps[si]
si = si + 1 # advance wheel index
if si > 92159: si = 0
sieve[nxt] = (s, si) # next non-marked multiple of a prime
nni = ni + 1 # advance wheel index
if nni > 92159: nni = 0
n += gaps[ni]; ni = nni # advance on the wheel
for p, pi in wheel_prime_pairs(): yield p # strip out indexes
 

[edit] R

This code is vectorised, so it runs quickly for n < one million. The allocation of the primes vector means memory usage is too high for n much larger than that.
sieve <- function(n)
{
n <- as.integer(n)
if(n > 1e6) stop("n too large")
primes <- rep(TRUE, n)
primes[1] <- FALSE
last.prime <- 2L
for(i in last.prime:floor(sqrt(n)))
{
primes[seq.int(2L*last.prime, n, last.prime)] <- FALSE
last.prime <- last.prime + min(which(primes[(last.prime+1):n]))
}
which(primes)
}
 
sieve(1000)

[edit] Racket

[edit] Imperative versions

Ugly imperative version:

#lang racket
 
(define (sieve n)
(define non-primes '())
(define primes '())
(for ([i (in-range 2 (add1 n))])
(unless (member i non-primes)
(set! primes (cons i primes))
(for ([j (in-range (* i i) (add1 n) i)])
(set! non-primes (cons j non-primes)))))
(reverse primes))
 
(sieve 100)

A little nicer, but still imperative:

#lang racket
(define (sieve n)
(define primes (make-vector (add1 n) #t))
(for* ([i (in-range 2 (add1 n))]
#:when (vector-ref primes i)
[j (in-range (* i i) (add1 n) i)])
(vector-set! primes j #f))
(for/list ([n (in-range 2 (add1 n))]
#:when (vector-ref primes n))
n))
(sieve 100)

Imperative version using a bit vector:

#lang racket
(require data/bit-vector)
;; Returns a list of prime numbers up to natural number limit
(define (eratosthenes limit)
(define bv (make-bit-vector (+ limit 1) #f))
(bit-vector-set! bv 0 #t)
(bit-vector-set! bv 1 #t)
(for* ([i (in-range (sqrt limit))] #:unless (bit-vector-ref bv i)
[j (in-range (* 2 i) (bit-vector-length bv) i)])
(bit-vector-set! bv j #t))
 ;; translate to a list of primes
(for/list ([i (bit-vector-length bv)] #:unless (bit-vector-ref bv i)) i))
(eratosthenes 100)
 
Output:

'(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97)

[edit] Infinite list of primes

[edit] Using laziness

These examples use infinite lists (streams) to implement the sieve of Eratosthenes in a functional way, and producing all prime numbers. The following functions are used as a prefix for pieces of code that follow:

#lang lazy
(define (ints-from i d) (cons i (ints-from (+ i d) d)))
(define (after n l f)
(if (< (car l) n) (cons (car l) (after n (cdr l) f)) (f l)))
(define (diff l1 l2)
(let ([x1 (car l1)] [x2 (car l2)])
(cond [(< x1 x2) (cons x1 (diff (cdr l1) l2 ))]
[(> x1 x2) (diff l1 (cdr l2)) ]
[else (diff (cdr l1) (cdr l2)) ])))
(define (union l1 l2)  ; union of two lists
(let ([x1 (car l1)] [x2 (car l2)])
(cond [(< x1 x2) (cons x1 (union (cdr l1) l2 ))]
[(> x1 x2) (cons x2 (union l1 (cdr l2)))]
[else (cons x1 (union (cdr l1) (cdr l2)))])))
[edit] Basic sieve
(define (sieve l)
(define x (car l))
(cons x (sieve (diff (cdr l) (ints-from (+ x x) x)))))
(define primes (sieve (ints-from 2 1)))
(!! (take 25 primes))

Runs at ~ n^2.1 empirically, for n <= 1500 primes produced.

[edit] With merged composites

Note that the first numbers, 2 and 3, are done manually outside of the sieve loop -- to ensure that the non-primes list is never empty, which simplifies the code since all functions assume non-empty infinite lists. This also means that the even numbers are ignored.

(define (sieve l non-primes)
(let ([x (car l)] [np (car non-primes)])
(cond [(= x np) (sieve (cdr l) (cdr non-primes))]  ; else x < np
[else (cons x (sieve (cdr l) (union (ints-from (* x x) (* 2 x))
non-primes)))])))
(define primes (cons 2 (cons 3 (sieve (ints-from 5 2) (ints-from 9 6)))))

Runs at ~ n^2.4 and worse, empirically, producing up to n=700 primes.

[edit] Basic sieve Optimized with postponed processing

Since a prime's multiples that count start from its square, we should only start removing them when we reach that square.

(define (sieve l prs)
(define p (car prs))
(define q (* p p))
(after q l (λ(t) (sieve (diff t (ints-from q p)) (cdr prs)))))
(define primes (cons 2 (sieve (ints-from 3 1) primes)))

Runs at ~ n^1.4 up to n=10,000. The initial 2 in the self-referential primes definition is needed to prevent a "black hole".

[edit] Merged composites Optimized with postponed processing

Since a prime's multiples that matter start from its square, we should only add them when we reach that square. Improves the time complexity to ~ n^1.5, tested producing up to n=6,000 primes.

(define (composites l q primes)
(after q l (λ(t) (let ([p (car primes)] [r (cadr primes)])
(composites (union t (ints-from q (* 2 p))) ; q = p*p
(* r r) (cdr primes))))))
(define primes (cons 2 (cons 3 (diff (ints-from 5 2)
(composites (ints-from 9 6) 25 (cddr primes))))))

The processing starts from odds only, ignoring all evens above 2, so effectively it is employing the simplest kind of a wheel.

[edit] Implementation of Richard Bird's algorithm

Appears in M.O'Neill's paper. Achieves on its own the proper postponement that is specifically arranged for in the version above (with after), and is yet more efficient, because it folds to the right and so builds the right-leaning structure of merges at run time, where the more frequently-producing streams of multiples appear higher in that structure, so the composite numbers produced by them have less merge nodes to percolate through:

(define primes
(cons 2 (diff (ints-from 3 1)
(foldr (λ(p r) (define q (* p p))
(cons q (union (ints-from (+ q p) p) r)))
'() primes))))

[edit] Using threads and channels

Same algorithm as "merged composites" above (without the optimization), but now using threads and channels to produce a channel of all prime numbers (similar to newsqueak). The macro at the top is a convenient wrapper around definitions of channels using a thread that feeds them.

#lang racket
(define-syntax (define-thread-loop stx)
(syntax-case stx ()
[(_ (name . args) expr ...)
(with-syntax ([out! (datum->syntax stx 'out!)])
#'(define (name . args)
(define out (make-channel))
(define (out! x) (channel-put out x))
(thread (λ() (let loop () expr ... (loop))))
out))]))
(define-thread-loop (ints-from i d) (out! i) (set! i (+ i d)))
(define-thread-loop (merge c1 c2)
(let loop ([x1 (channel-get c1)] [x2 (channel-get c2)])
(cond [(< x1 x2) (out! x1) (loop (channel-get c1) x2)]
[(> x1 x2) (out! x2) (loop x1 (channel-get c2))]
[else (out! x1) (loop (channel-get c1) (channel-get c2))])))
(define-thread-loop (sieve l non-primes)
(let loop ([x (channel-get l)] [np (channel-get non-primes)])
(cond [(< np x) (loop x (channel-get non-primes))]
[(= np x) (loop (channel-get l) (channel-get non-primes))]
[else (out! x)
(set! non-primes (merge (ints-from (* x x) x) non-primes))])))
(define-thread-loop (cons x l)
(out! x) (let loop () (out! (channel-get l)) (loop)))
(define primes (cons 2 (sieve (ints-from 3 2) (ints-from 2 2))))
(for/list ([i 25] [x (in-producer channel-get eof primes)]) x)

[edit] Using generators

Yet another variation of the same algorithm as above, this time using generators.

#lang racket
(require racket/generator)
(define (ints-from i d)
(generator () (let loop ([i i]) (yield i) (loop (+ i d)))))
(define (merge g1 g2)
(generator ()
(let loop ([x1 (g1)] [x2 (g2)])
(cond [(< x1 x2) (yield x1) (loop (g1) x2)]
[(> x1 x2) (yield x2) (loop x1 (g2))]
[else (yield x1) (loop (g1) (g2))]))))
(define (sieve l non-primes)
(generator ()
(let loop ([x (l)] [np (non-primes)])
(cond [(< np x) (loop x (non-primes))]
[(= np x) (loop (l) (non-primes))]
[else (yield x)
(set! non-primes (merge (ints-from (* x x) x) non-primes))
(loop (l) (non-primes))]))))
(define (cons x l) (generator () (yield x) (let loop () (yield (l)) (loop))))
(define primes (cons 2 (sieve (ints-from 3 2) (ints-from 2 2))))
(for/list ([i 25] [x (in-producer primes)]) x)

[edit] REXX

[edit] no wheel version

The first three REXX versions make use of a sparse stemmed array:   [@.].
As the stemmed array gets heavily populated, the number of entries may slow down the REXX interpreter substantially,
depending upon the efficacy of the hashing technique being used for REXX variables (setting/retrieving).

/*REXX program generates primes via the sieve of Eratosthenes algorithm.*/
parse arg H .; if H=='' then H=200 /*was the high limit specified? */
w=length(H); @prime=right('prime',20) /*W is used for formatting output*/
@.=. /*assume all numbers are prime. */
#=0 /*number of primes found so far. */
do j=2 for H-1 /*all integers up to H inclusive.*/
if @.j=='' then iterate /*Composite? Then skip this num.*/
#=#+1 /*bump the prime number counter. */
say @prime right(#,w) " ───► " right(j,w) /*show the prime.*/
do m=j*j to H by j; @.m=; end /*strike all multiples as ¬ prime*/
end /*j*/ /* ─── */
/*stick a fork in it, we're done.*/
say; say right(#,w+length(@prime)+1) 'primes found.'

output when using the input default of:   200

               prime   1  ───►    2
               prime   2  ───►    3
               prime   3  ───►    5
               prime   4  ───►    7
               prime   5  ───►   11
               prime   6  ───►   13
               prime   7  ───►   17
               prime   8  ───►   19
               prime   9  ───►   23
               prime  10  ───►   29
               prime  11  ───►   31
               prime  12  ───►   37
               prime  13  ───►   41
               prime  14  ───►   43
               prime  15  ───►   47
               prime  16  ───►   53
               prime  17  ───►   59
               prime  18  ───►   61
               prime  19  ───►   67
               prime  20  ───►   71
               prime  21  ───►   73
               prime  22  ───►   79
               prime  23  ───►   83
               prime  24  ───►   89
               prime  25  ───►   97
               prime  26  ───►  101
               prime  27  ───►  103
               prime  28  ───►  107
               prime  29  ───►  109
               prime  30  ───►  113
               prime  31  ───►  127
               prime  32  ───►  131
               prime  33  ───►  137
               prime  34  ───►  139
               prime  35  ───►  149
               prime  36  ───►  151
               prime  37  ───►  157
               prime  38  ───►  163
               prime  39  ───►  167
               prime  40  ───►  173
               prime  41  ───►  179
               prime  42  ───►  181
               prime  43  ───►  191
               prime  44  ───►  193
               prime  45  ───►  197
               prime  46  ───►  199

                      46 primes found.

[edit] wheel version, optional prime list suppression

This version skips striking the even numbers   (as being not prime),   2   is handled as a special case.

Also supported is the suppression of listing the primes if the H (high limit) is negative.
Also added is a final message indicating the number of primes found.

/*REXX pgm gens primes via a  wheeled  sieve of Eratosthenes  algorithm.*/
parse arg H .; if H=='' then H=200 /*let the highest # be specified.*/
tell=h>0; H=abs(H); w=length(H) /*neg H suppresses prime listing.*/
if 2<=H & tell then say right(1,w+20)'st prime ───► ' right(2,w)
#= w<=H /*number of primes found so far. */
@.=. /*assume all numbers are prime. */
!=0 /*skips top part of sieve marking*/
do j=3 by 2 for (H-2)%2 /*odd integers up to H inclusive.*/
if @.j=='' then iterate /*composite? Then skip this num.*/
#=#+1 /*bump the prime number counter. */
if tell then say right(#,w+20)th(#) 'prime ───► ' right(j,w)
if ! then iterate /*should the top part be skipped?*/
jj=j*j /*compute the square of J. __ */
if jj>H then !=1 /*indicate skipping if j > √ H.*/
do m=jj to H by j+j; @.m=; end /*strike odd multiples as ¬ prime*/
end /*j*/ /* ─── */
say
say right(#,w+20) 'prime's(#) "found."
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────one─liner subroutines────────────────────────────────────────*/
s: if arg(1)==1 then return arg(3); return word(arg(2) 's',1) /*pluralizer.*/
th: procedure; parse arg x; x=abs(x); return word('th st nd rd',1+x//10*(x//100%10\==1)*(x//10<4))

output when using the input default of:   200

                      1st prime   ───►    2
                      2nd prime   ───►    3
                      3rd prime   ───►    5
                      4th prime   ───►    7
                      5th prime   ───►   11
                      6th prime   ───►   13
                      7th prime   ───►   17
                      8th prime   ───►   19
                      9th prime   ───►   23
                     10th prime   ───►   29
                     11th prime   ───►   31
                     12th prime   ───►   37
                     13th prime   ───►   41
                     14th prime   ───►   43
                     15th prime   ───►   47
                     16th prime   ───►   53
                     17th prime   ───►   59
                     18th prime   ───►   61
                     19th prime   ───►   67
                     20th prime   ───►   71
                     21st prime   ───►   73
                     22nd prime   ───►   79
                     23rd prime   ───►   83
                     24th prime   ───►   89
                     25th prime   ───►   97
                     26th prime   ───►  101
                     27th prime   ───►  103
                     28th prime   ───►  107
                     29th prime   ───►  109
                     30th prime   ───►  113
                     31st prime   ───►  127
                     32nd prime   ───►  131
                     33rd prime   ───►  137
                     34th prime   ───►  139
                     35th prime   ───►  149
                     36th prime   ───►  151
                     37th prime   ───►  157
                     38th prime   ───►  163
                     39th prime   ───►  167
                     40th prime   ───►  173
                     41st prime   ───►  179
                     42nd prime   ───►  181
                     43rd prime   ───►  191
                     44th prime   ───►  193
                     45th prime   ───►  197
                     46th prime   ───►  199

                       46 primes found.

output when using the input of:   -1000

                       168 primes found.

output when using the input of:   -10000

                       1229 primes found.

output when using the input of:   -100000

                        9592 primes found.

output when using the input of:   -1000000

                        78498 primes found.

output when using the input of:   -10000000

                        664579 primes found.

output when using the input of:   -100000000

    16 +++       @.m=
Error 5 running "C:\sieve_of_Eratosthenes.rex", line 16: System resources exhausted

The above (using Regina 3.82 under Windows/XP) shows one of the weaknesses of this implementation of the Sieve of Eratosthenes:   it must keep an array of all (if not most) values which is used to strike out composite numbers.

The   System resources exhausted   error can be postponed by implementing further optimizations (expanding the wheel with low primes).

[edit] wheel version

This version skips striking the even numbers   (as being not prime).
It also uses a short-circuit test for striking out composites ≤ √target.

/*REXX pgm gens primes via a  wheeled  sieve of Eratosthenes  algorithm.*/
parse arg H .; if H=='' then H=200 /*high# can be specified on C.L. */
w=length(H); @prime=right('prime', 20) /*W is used for formatting output*/
if 2<=H then say @prime right(1,w) " ───► " right(2,w)
#= 2<=H /*number of primes found so far. */
@.=. /*assume all numbers are prime. */
!=0 /*skips top part of sieve marking*/
do j=3 by 2 for (H-2)%2 /*odd integers up to H inclusive.*/
if @.j=='' then iterate /*composite? Then skip this num.*/
#=#+1 /*bump the prime number counter. */
say @prime right(#,w) " ───► " right(j,w) /*show the prime.*/
if ! then iterate /*should the top part be skipped?*/
jj=j*j /*compute the square of J. __ */
if jj>H then !=1 /*indicate skipping if j > √ H.*/
do m=jj to H by j+j; @.m=; end /*strike odd multiples as ¬ prime*/
end /*j*/ /* ─── */
say /*stick a fork in it, we're done.*/
say right(#, w+length(@prime)+1) 'primes found.'

output is identical to the first (non-wheel) version;   program execution is over twice as fast.
The addition of the short-circuit test (variable !) makes it about another 20% faster.

[edit] Wheel Version restructured

/*REXX program generates primes via sieve of Eratosthenes algorithm.
* 21.07.2012 Walter Pachl derived from above Rexx version
* avoid symbols @ and # (not supported by ooRexx)
* avoid negations (think positive)
**********************************************************************/

highest=200 /*define highest number to use. */
is_prime.=1 /*assume all numbers are prime. */
w=length(highest) /*width of the biggest number, */
/* it's used for aligned output.*/
Do j=3 To highest By 2, /*strike multiples of odd ints. */
While j*j<=highest /* up to sqrt(highest) */
If is_prime.j Then Do
Do jm=j*3 To highest By j+j /*start with next odd mult. of J */
is_prime.jm=0 /*mark odd mult. of J not prime. */
End
End
End
np=0 /*number of primes shown */
Call tell 2
Do n=3 To highest By 2 /*list all the primes found. */
If is_prime.n Then Do
Call tell n
End
End
Exit
tell: Parse Arg prime
np=np+1
Say ' prime number' right(np,w) " --> " right(prime,w)
Return

output is identical to the above versions.

[edit] Ruby

eratosthenes starts with nums = [nil, nil, 2, 3, 4, 5, ..., n], then marks ( the nil setting ) multiples of 2, 3, 5, 7, ... there, then returns all non-nil numbers which are the primes.

def eratosthenes(n)
nums = [nil, nil, *2..n]
(2..Math.sqrt(n)).each do |i|
(i**2..n).step(i){