Almost prime
You are encouraged to solve this task according to the task description, using any language you may know.
A k-Almost-prime is a natural number that is the product of (possibly identical) primes.
- Example
1-almost-primes, where , are the prime numbers themselves.
2-almost-primes, where , are the semiprimes.
- Task
Write a function/method/subroutine/... that generates k-almost primes and use it to create a table here of the first ten members of k-Almost primes for .
- Related tasks
11l
F k_prime(k, =n)
V f = 0
V p = 2
L f < k & p * p <= n
L n % p == 0
n /= p
f++
p++
R f + (I n > 1 {1} E 0) == k
F primes(k, n)
V i = 2
[Int] list
L list.len < n
I k_prime(k, i)
list [+]= i
i++
R list
L(k) 1..5
print(‘k = ’k‘: ’primes(k, 10))- Output:
k = 1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] k = 2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] k = 3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] k = 4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] k = 5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Action!
BYTE FUNC IsAlmostPrime(INT num BYTE k)
INT f,p,v
f=0 p=2 v=num
WHILE f<k AND p*p<=num
DO
WHILE v MOD p=0
DO
v==/p f==+1
OD
p==+1
OD
IF v>1 THEN
f==+1
FI
IF f=k THEN
RETURN (1)
FI
RETURN (0)
PROC Main()
BYTE count,k
INT i
FOR k=1 TO 5
DO
PrintF("k=%B:",k)
count=0 i=2
WHILE count<10
DO
IF IsAlmostPrime(i,k) THEN
PrintF(" %I",i)
count==+1
FI
i==+1
OD
PutE()
OD
RETURN- Output:
Screenshot from Atari 8-bit computer
k=1: 2 3 5 7 11 13 17 19 23 29 k=2: 4 6 9 10 14 15 21 22 25 26 k=3: 8 12 18 20 27 28 30 42 44 45 k=4: 16 24 36 40 54 56 60 81 84 88 k=5: 32 48 72 80 108 112 120 162 168 176
Ada
This imports the package Prime_Numbers from Prime decomposition#Ada.
with Prime_Numbers, Ada.Text_IO;
procedure Test_Kth_Prime is
package Integer_Numbers is new
Prime_Numbers (Natural, 0, 1, 2);
use Integer_Numbers;
Out_Length: constant Positive := 10; -- 10 k-th almost primes
N: Positive; -- the "current number" to be checked
begin
for K in 1 .. 5 loop
Ada.Text_IO.Put("K =" & Integer'Image(K) &": ");
N := 2;
for I in 1 .. Out_Length loop
while Decompose(N)'Length /= K loop
N := N + 1;
end loop; -- now N is Kth almost prime;
Ada.Text_IO.Put(Integer'Image(Integer(N)));
N := N + 1;
end loop;
Ada.Text_IO.New_Line;
end loop;
end Test_Kth_Prime;
- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
ALGOL 68
Worth noticing is the n(...)(...) picture in the printf and the WHILE ... DO SKIP OD idiom which is quite common in ALgol 68.
BEGIN
INT examples=10, classes=5;
MODE SEMIPRIME = STRUCT ([examples]INT data, INT count);
[classes]SEMIPRIME semi primes;
PROC num facs = (INT n) INT :
COMMENT
Return number of not necessarily distinct prime factors of n.
Not very efficient for large n ...
COMMENT
BEGIN
INT tf := 2, residue := n, count := 1;
WHILE tf < residue DO
INT remainder = residue MOD tf;
( remainder = 0 | count +:= 1; residue %:= tf | tf +:= 1 )
OD;
count
END;
PROC update table = (REF []SEMIPRIME table, INT i) BOOL :
COMMENT
Add i to the appropriate row of the table, if any, unless that row
is already full. Return a BOOL which is TRUE when all of the table
is full.
COMMENT
BEGIN
INT k := num facs(i);
IF k <= classes
THEN
INT c = 1 + count OF table[k];
( c <= examples | (data OF table[k])[c] := i; count OF table[k] := c )
FI;
INT sum := 0;
FOR i TO classes DO sum +:= count OF table[i] OD;
sum < classes * examples
END;
FOR i TO classes DO count OF semi primes[i] := 0 OD;
FOR i FROM 2 WHILE update table (semi primes, i) DO SKIP OD;
FOR i TO classes
DO
printf (($"k = ", d, ":", n(examples)(xg(0))l$, i, data OF semi primes[i]))
OD
END- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
ALGOL-M
begin
integer function mod(a, b);
integer a, b;
mod := a-(a/b)*b;
integer function kprime(n, k);
integer n, k;
begin
integer p, f;
f := 0;
p := 2;
while f < k and p*p <= n do
begin
while mod(n,p) = 0 do
begin
n := n / p;
f := f + 1;
end;
p := p + 1;
end;
if n > 1 then f := f + 1;
if f = k then kprime := 1 else kprime := 0;
end;
integer i, c, k;
for k := 1 step 1 until 5 do
begin
write("k =");
writeon(k);
writeon(": ");
c := 0;
i := 2;
while c < 10 do
begin
if kprime(i, k) <> 0 then
begin
writeon(i);
c := c + 1;
end;
i := i + 1;
end;
end;
end- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
ALGOL W
with tweaks to the factorisation routine.
begin
logical procedure kPrime( integer value nv, k ) ;
begin
integer p, f, n;
n := nv;
f := 0;
while f <= k and not odd( n ) do begin
n := n div 2;
f := f + 1
end while_not_odd_n ;
p := 3;
while f <= k and p * p <= n do begin
while n rem p = 0 do begin
n := n div p;
f := f + 1
end while_n_rem_p_eq_0 ;
p := p + 2
end while_f_le_k_and_p_is_a_factor ;
if n > 1 then f := f + 1;
f = k
end kPrime ;
begin
for k := 1 until 5 do begin
integer c, i;
write( i_w := 1, s_w := 0, "k = ", k , ": " );
c := 0;
i := 2;
while c < 10 do begin
if kPrime( i, k ) then begin
writeon( i_w := 3, s_w := 0, " ", i );
c := c + 1
end if_kPrime_i_k ;
i := i + 1
end while_c_lt_10
end for_k
end
end.- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
APL
Works in Dyalog APL
f←{↑r⊣⍵∘{r,∘⊂←⍺↑∪{⍵[⍋⍵]},f∘.×⍵}⍣(⍺-1)⊃r←⊂f←pco¨⍳⍵}
- Output:
5 f 10 2 3 5 7 11 13 17 19 23 29 4 6 9 10 14 15 21 22 25 26 8 12 18 20 27 28 30 42 44 45 16 24 36 40 54 56 60 81 84 88 32 48 72 80 108 112 120 162 168 176
ARM Assembly
/* ARM assembly Raspberry PI */
/* program kprime.s */
/************************************/
/* Constantes */
/************************************/
.equ STDOUT, 1 @ Linux output console
.equ EXIT, 1 @ Linux syscall
.equ WRITE, 4 @ Linux syscall
.equ MAXI, 10
.equ MAXIK, 5
/*********************************/
/* Initialized data */
/*********************************/
.data
sMessDeb: .ascii "k="
sMessValeurDeb: .fill 11, 1, ' ' @ size => 11
sMessResult: .ascii " "
sMessValeur: .fill 11, 1, ' ' @ size => 11
szCarriageReturn: .asciz "\n"
/*********************************/
/* UnInitialized data */
/*********************************/
.bss
/*********************************/
/* code section */
/*********************************/
.text
.global main
main: @ entry of program
mov r3,#1 @ k
1: @ start loop k
mov r0,r3
ldr r1,iAdrsMessValeurDeb
bl conversion10 @ call conversion decimal
ldr r0,iAdrsMessValeurDeb
mov r1,#':'
strb r1,[r0,#2] @ write : after k value
mov r1,#0
strb r1,[r0,#3] @ final zéro
ldr r0,iAdrsMessDeb
bl affichageMess @ display message
mov r4,#2 @ n
mov r5,#0 @ result counter
2: @ start loop n
mov r0,r4
mov r1,r3
bl kprime @ is kprine ?
cmp r0,#0
beq 3f @ no
mov r0,r4
ldr r1,iAdrsMessValeur
bl conversion10 @ call conversion decimal
ldr r0,iAdrsMessValeur
mov r1,#0
strb r1,[r0,#4] @ final zéro
ldr r0,iAdrsMessResult
bl affichageMess @ display message
add r5,#1 @ increment counter
3:
add r4,#1 @ increment n
cmp r5,#MAXI @ maxi ?
blt 2b @ no -> loop
ldr r0,iAdrszCarriageReturn
bl affichageMess @ display carriage return
add r3,#1 @ increment k
cmp r3,#MAXIK @ maxi ?
ble 1b @ no -> loop
100: @ standard end of the program
mov r0, #0 @ return code
mov r7, #EXIT @ request to exit program
svc #0 @ perform the system call
iAdrsMessValeur: .int sMessValeur
iAdrszCarriageReturn: .int szCarriageReturn
iAdrsMessResult: .int sMessResult
iAdrsMessValeurDeb: .int sMessValeurDeb
iAdrsMessDeb: .int sMessDeb
/******************************************************************/
/* compute kprime (n,k) */
/******************************************************************/
/* r0 contains n */
/* r1 contains k */
kprime:
push {r1-r7,lr} @ save registers
mov r5,r0 @ save n
mov r7,r1 @ save k
mov r4,#0 @ counter product
mov r1,#2 @ divisor
1: @ start loop
cmp r4,r7 @ counter >= k
bge 4f @ yes -> end
mul r6,r1,r1 @ compute product
cmp r6,r5 @ > n
bgt 4f @ yes -> end
2: @ start loop division
mov r0,r5 @ dividende
bl division @ by r1
cmp r3,#0 @ remainder = 0 ?
bne 3f @ no
mov r5,r2 @ yes -> n = n / r1
add r4,#1 @ increment counter
b 2b @ and loop
3:
add r1,#1 @ increment divisor
b 1b @ and loop
4: @ end compute
cmp r5,#1 @ n > 1
addgt r4,#1 @ yes increment counter
cmp r4,r7 @ counter = k ?
movne r0,#0 @ no -> no kprime
moveq r0,#1 @ yes -> kprime
100:
pop {r1-r7,lr} @ restaur registers
bx lr @return
/******************************************************************/
/* display text with size calculation */
/******************************************************************/
/* r0 contains the address of the message */
affichageMess:
push {r0,r1,r2,r7,lr} @ save registres
mov r2,#0 @ counter length
1: @ loop length calculation
ldrb r1,[r0,r2] @ read octet start position + index
cmp r1,#0 @ if 0 its over
addne r2,r2,#1 @ else add 1 in the length
bne 1b @ and loop
@ so here r2 contains the length of the message
mov r1,r0 @ address message in r1
mov r0,#STDOUT @ code to write to the standard output Linux
mov r7, #WRITE @ code call system "write"
svc #0 @ call systeme
pop {r0,r1,r2,r7,lr} @ restaur des 2 registres */
bx lr @ return
/******************************************************************/
/* Converting a register to a decimal unsigned */
/******************************************************************/
/* r0 contains value and r1 address area */
/* r0 return size of result (no zero final in area) */
/* area size => 11 bytes */
.equ LGZONECAL, 10
conversion10:
push {r1-r4,lr} @ save registers
mov r3,r1
mov r2,#LGZONECAL
1: @ start loop
bl divisionpar10U @ unsigned r0 <- dividende. quotient ->r0 reste -> r1
add r1,#48 @ digit
strb r1,[r3,r2] @ store digit on area
cmp r0,#0 @ stop if quotient = 0
subne r2,#1 @ else previous position
bne 1b @ and loop
@ and move digit from left of area
mov r4,#0
2:
ldrb r1,[r3,r2]
strb r1,[r3,r4]
add r2,#1
add r4,#1
cmp r2,#LGZONECAL
ble 2b
@ and move spaces in end on area
mov r0,r4 @ result length
mov r1,#' ' @ space
3:
strb r1,[r3,r4] @ store space in area
add r4,#1 @ next position
cmp r4,#LGZONECAL
ble 3b @ loop if r4 <= area size
100:
pop {r1-r4,lr} @ restaur registres
bx lr @return
/***************************************************/
/* division par 10 unsigned */
/***************************************************/
/* r0 dividende */
/* r0 quotient */
/* r1 remainder */
divisionpar10U:
push {r2,r3,r4, lr}
mov r4,r0 @ save value
ldr r3,iMagicNumber @ r3 <- magic_number raspberry 1 2
umull r1, r2, r3, r0 @ r1<- Lower32Bits(r1*r0) r2<- Upper32Bits(r1*r0)
mov r0, r2, LSR #3 @ r2 <- r2 >> shift 3
add r2,r0,r0, lsl #2 @ r2 <- r0 * 5
sub r1,r4,r2, lsl #1 @ r1 <- r4 - (r2 * 2) = r4 - (r0 * 10)
pop {r2,r3,r4,lr}
bx lr @ leave function
iMagicNumber: .int 0xCCCCCCCD
/***************************************************/
/* integer division unsigned */
/***************************************************/
division:
/* r0 contains dividend */
/* r1 contains divisor */
/* r2 returns quotient */
/* r3 returns remainder */
push {r4, lr}
mov r2, #0 @ init quotient
mov r3, #0 @ init remainder
mov r4, #32 @ init counter bits
b 2f
1: @ loop
movs r0, r0, LSL #1 @ r0 <- r0 << 1 updating cpsr (sets C if 31st bit of r0 was 1)
adc r3, r3, r3 @ r3 <- r3 + r3 + C. This is equivalent to r3 ? (r3 << 1) + C
cmp r3, r1 @ compute r3 - r1 and update cpsr
subhs r3, r3, r1 @ if r3 >= r1 (C=1) then r3 <- r3 - r1
adc r2, r2, r2 @ r2 <- r2 + r2 + C. This is equivalent to r2 <- (r2 << 1) + C
2:
subs r4, r4, #1 @ r4 <- r4 - 1
bpl 1b @ if r4 >= 0 (N=0) then loop
pop {r4, lr}
bx lrOutput:
k=1 : 2 3 5 7 11 13 17 19 23 29 k=2 : 4 6 9 10 14 15 21 22 25 26 k=3 : 8 12 18 20 27 28 30 42 44 45 k=4 : 16 24 36 40 54 56 60 81 84 88 k=5 : 32 48 72 80 108 112 120 162 168 176
Arturo
almostPrime: function [k, listLen][
result: new []
test: 2
c: 0
while [c < listLen][
i: 2
m: 0
n: test
while [i =< n][
if? zero? n % i [
n: n / i
m: m + 1
]
else -> i: i + 1
]
if m = k [
'result ++ test
c: c + 1
]
test: test + 1
]
return result
]
loop 1..5 'x ->
print ["k:" x "=>" almostPrime x 10]
- Output:
k: 1 => [2 3 5 7 11 13 17 19 23 29] k: 2 => [4 6 9 10 14 15 21 22 25 26] k: 3 => [8 12 18 20 27 28 30 42 44 45] k: 4 => [16 24 36 40 54 56 60 81 84 88] k: 5 => [32 48 72 80 108 112 120 162 168 176]
AutoHotkey
Translation of the C Version
kprime(n,k) {
p:=2, f:=0
while( (f<k) && (p*p<=n) ) {
while ( 0==mod(n,p) ) {
n/=p
f++
}
p++
}
return f + (n>1) == k
}
k:=1, results:=""
while( k<=5 ) {
i:=2, c:=0, results:=results "k =" k ":"
while( c<10 ) {
if (kprime(i,k)) {
results:=results " " i
c++
}
i++
}
results:=results "`n"
k++
}
MsgBox % results
Output (Msgbox):
k =1: 2 3 5 7 11 13 17 19 23 29 k =2: 4 6 9 10 14 15 21 22 25 26 k =3: 8 12 18 20 27 28 30 42 44 45 k =4: 16 24 36 40 54 56 60 81 84 88 k =5: 32 48 72 80 108 112 120 162 168 176
AWK
# syntax: GAWK -f ALMOST_PRIME.AWK
BEGIN {
for (k=1; k<=5; k++) {
printf("%d:",k)
c = 0
i = 1
while (c < 10) {
if (kprime(++i,k)) {
printf(" %d",i)
c++
}
}
printf("\n")
}
exit(0)
}
function kprime(n,k, f,p) {
for (p=2; f<k && p*p<=n; p++) {
while (n % p == 0) {
n /= p
f++
}
}
return(f + (n > 1) == k)
}
Output:
1: 2 3 5 7 11 13 17 19 23 29 2: 4 6 9 10 14 15 21 22 25 26 3: 8 12 18 20 27 28 30 42 44 45 4: 16 24 36 40 54 56 60 81 84 88 5: 32 48 72 80 108 112 120 162 168 176
BASIC
10 DEFINT A-Z
20 FOR K=1 TO 5
30 PRINT USING "K = #:";K;
40 I=2: C=0
50 F=0: P=2: N=I
60 IF F >= K OR P*P > N THEN 100
70 IF N MOD P = 0 THEN N = N/P: F = F+1: GOTO 70
80 P = P+1
90 GOTO 60
100 IF N > 1 THEN F = F+1
110 IF F = K THEN C = C+1: PRINT USING " ###";I;
120 I = I+1
130 IF C < 10 THEN 50
140 PRINT
150 NEXT K
- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
ASIC
ASIC has both FOR and WHILE loops, but it had better not go out from the loop. So, in the subroutine CHECKKPRIME they are simulated by the constructs with GOTO statements.
REM Almost prime
FOR K = 1 TO 5
S$ = STR$(K)
S$ = LTRIM$(S$)
S$ = "k = " + S$
S$ = S$ + ":"
PRINT S$;
I = 2
C = 0
WHILE C < 10
AN = I
GOSUB CHECKKPRIME:
IF ISKPRIME <> 0 THEN
PRINT I;
C = C + 1
ENDIF
I = I + 1
WEND
PRINT
NEXT K
END
CHECKKPRIME:
REM Check if N (AN) is a K prime (result: ISKPRIME)
F = 0
J = 2
LOOPFOR:
ANMODJ = AN MOD J
LOOPWHILE:
IF ANMODJ <> 0 THEN AFTERWHILE:
IF F = K THEN FEQK:
F = F + 1
AN = AN / J
ANMODJ = AN MOD J
GOTO LOOPWHILE:
AFTERWHILE:
J = J + 1
IF J <= AN THEN LOOPFOR:
IF F = K THEN
ISKPRIME = -1
ELSE
ISKPRIME = 0
ENDIF
RETURN
FEQK:
ISKPRIME = 0
RETURN
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
BASIC256
function kPrime(n, k)
f = 0
for i = 2 to n
while n mod i = 0
if f = k then return False
f += 1
n /= i
end while
next i
return f = k
end function
for k = 1 to 5
print "k = "; k; " :";
i = 2
c = 0
while c < 10
if kPrime(i, k) then
print rjust (string(i), 4);
c += 1
end if
i += 1
end while
print
next k
end
Chipmunk Basic
10 'Almost prime
20 FOR k = 1 TO 5
30 PRINT "k = "; k; ":";
40 LET i = 2
50 LET c = 0
60 WHILE c < 10
70 LET an = i: GOSUB 150
80 IF iskprime <> 0 THEN PRINT USING " ###"; i; : LET c = c + 1
90 LET i = i + 1
100 WEND
110 PRINT
120 NEXT k
130 END
140 ' Check if n (AN) is a k (K) prime
150 LET f = 0
160 FOR j = 2 TO an
170 WHILE an MOD j = 0
180 IF f = k THEN LET iskprime = 0: RETURN
190 LET f = f + 1
200 LET an = INT(an / j)
210 WEND
220 NEXT j
230 LET iskprime = (f = k)
240 RETURN
Craft Basic
for k = 1 to 5
print "k = ", k, ": ",
let e = 2
let c = 0
do
if c < 10 then
let n = e
gosub kprime
if r then
print tab, e,
let c = c + 1
endif
let e = e + 1
endif
loop c < 10
print
next k
end
sub kprime
let f = 0
for i = 2 to n
do
if n mod i = 0 then
if f = k then
let r = 0
return
endif
let f = f + 1
let n = n / i
wait
endif
loop n mod i = 0
next i
let r = f = k
return
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
FreeBASIC
' FB 1.05.0 Win64
Function kPrime(n As Integer, k As Integer) As Boolean
Dim f As Integer = 0
For i As Integer = 2 To n
While n Mod i = 0
If f = k Then Return false
f += 1
n \= i
Wend
Next
Return f = k
End Function
Dim As Integer i, c, k
For k = 1 To 5
Print "k = "; k; " : ";
i = 2
c = 0
While c < 10
If kPrime(i, k) Then
Print Using "### "; i;
c += 1
End If
i += 1
Wend
Print
Next
Print
Print "Press any key to quit"
Sleep
- Output:
k = 1 : 2 3 5 7 11 13 17 19 23 29 k = 2 : 4 6 9 10 14 15 21 22 25 26 k = 3 : 8 12 18 20 27 28 30 42 44 45 k = 4 : 16 24 36 40 54 56 60 81 84 88 k = 5 : 32 48 72 80 108 112 120 162 168 176
Gambas
Public Sub Main()
Dim i As Integer, c As Integer, k As Integer
For k = 1 To 5
Print "k = "; k; " : ";
i = 2
c = 0
While c < 10
If kPrime(i, k) Then
Print Format$(Str$(i), "### ");
c += 1
End If
i += 1
Wend
Print
Next
End
Function kPrime(n As Integer, k As Integer) As Boolean
Dim f As Integer = 0
For i As Integer = 2 To n
While n Mod i = 0
If f = k Then Return False
f += 1
n \= i
Wend
Next
Return f = k
End Function
- Output:
Same as FreeBASIC entry.
GW-BASIC
10 'Almost prime
20 FOR K% = 1 TO 5
30 PRINT "k = "; K%; ":";
40 LET I% = 2
50 LET C% = 0
60 WHILE C% < 10
70 LET AN% = I%: GOSUB 1000
80 IF ISKPRIME <> 0 THEN PRINT USING " ###"; I%;: LET C% = C% + 1
90 LET I% = I% + 1
100 WEND
110 PRINT
120 NEXT K%
130 END
995 ' Check if n (AN%) is a k (K%) prime
1000 LET F% = 0
1010 FOR J% = 2 TO AN%
1020 WHILE AN% MOD J% = 0
1030 IF F% = K% THEN LET ISKPRIME = 0: RETURN
1040 LET F% = F% + 1
1050 LET AN% = AN% \ J%
1060 WEND
1070 NEXT J%
1080 LET ISKPRIME = (F% = K%)
1090 RETURN
- Output:
k = 1 : 2 3 5 7 11 13 17 19 23 29 k = 2 : 4 6 9 10 14 15 21 22 25 26 k = 3 : 8 12 18 20 27 28 30 42 44 45 k = 4 : 16 24 36 40 54 56 60 81 84 88 k = 5 : 32 48 72 80 108 112 120 162 168 176
Liberty BASIC
' Almost prime
for k = 1 to 5
print "k = "; k; ":";
i = 2
c = 0
while c < 10
if kPrime(i, k) then
print " "; using("###", i);
c = c + 1
end if
i = i + 1
wend
print
next k
end
function kPrime(n, k)
f = 0
for i = 2 to n
while n mod i = 0
if f = k then kPrime = 0: exit function
f = f + 1
n = int(n / i)
wend
next i
kPrime = abs(f = k)
end function- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Nascom BASIC
10 REM Almost prime
20 FOR K=1 TO 5
30 PRINT "k =";STR$(K);":";
40 I=2
50 C=0
60 IF C>=10 THEN 110
70 AN=I:GOSUB 1000
80 IF ISKPRIME=0 THEN 90
82 REM Print I in 4 fields
84 S$=STR$(I)
86 PRINT SPC(4-LEN(S$));S$;
88 C=C+1
90 I=I+1
100 GOTO 60
110 PRINT
120 NEXT K
130 END
995 REM Check if N (AN) is a K prime
1000 F=0
1010 FOR J=2 TO AN
1020 IF INT(AN/J)*J<>AN THEN 1070
1030 IF F=K THEN ISKPRIME=0:RETURN
1040 F=F+1
1050 AN=INT(AN/J)
1060 GOTO 1020
1070 NEXT J
1080 ISKPRIME=(F=K)
1090 RETURN
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
PureBasic
EnableExplicit
Procedure.b kprime(n.i, k.i)
Define p.i = 2,
f.i = 0
While f < k And p*p <= n
While n % p = 0
n / p
f + 1
Wend
p + 1
Wend
ProcedureReturn Bool(f + Bool(n > 1) = k)
EndProcedure
;___main____
If Not OpenConsole("Almost prime")
End -1
EndIf
Define i.i,
c.i,
k.i
For k = 1 To 5
Print("k = " + Str(k) + ":")
i = 2
c = 0
While c < 10
If kprime(i, k)
Print(RSet(Str(i),4))
c + 1
EndIf
i + 1
Wend
PrintN("")
Next
Input()
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Run BASIC
for k = 1 to 5
print "k = "; k; " :";
i = 2
c = 0
while c < 10
if kPrime(i, k) then
print " "; using("###", i);
c = c +1
end if
i = i +1
wend
print
next k
end
function kPrime(n, k)
f = 0
for i = 2 to n
while n mod i = 0
if f = k then kPrime = 0
f = f +1
n = int(n / i)
wend
next i
kPrime = abs(f = k)
end function
Tiny BASIC
REM Almost prime
LET K=1
10 IF K>5 THEN END
PRINT "k = ",K,":"
LET I=2
LET C=0
20 IF C>=10 THEN GOTO 40
LET N=I
GOSUB 500
IF P=0 THEN GOTO 30
PRINT I
LET C=C+1
30 LET I=I+1
GOTO 20
40 LET K=K+1
GOTO 10
REM Check if N is a K prime (result: P)
500 LET F=0
LET J=2
510 IF (N/J)*J<>N THEN GOTO 520
IF F=K THEN GOTO 530
LET F=F+1
LET N=N/J
GOTO 510
520 LET J=J+1
IF J<=N THEN GOTO 510
LET P=0
IF F=K THEN LET P=-1
RETURN
530 LET P=0
RETURN- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
True BASIC
FUNCTION iskprime(n, k)
! Check if n (AN) is a k (K) prime
LET f = 0
FOR j = 2 TO an
DO WHILE REMAINDER(an, j) = 0
IF f = k THEN LET iskprime = 0
LET f = f + 1
LET an = INT(an / j)
LOOP
NEXT j
IF (f = k) THEN LET iskprime = 1
END FUNCTION
!ALMOST prime
FOR k = 1 TO 5
PRINT "k = "; k; ":";
LET i = 2
LET c = 0
DO WHILE c < 10
LET an = i
IF iskprime(i,k) <> 0 THEN
PRINT USING " ###": i;
LET c = c + 1
END IF
LET i = i + 1
LOOP
PRINT
NEXT k
END
uBasic/4tH
Local(3)
For c@ = 1 To 5
Print "k = ";c@;": ";
b@=0
For a@ = 2 Step 1 While b@ < 10
If FUNC(_kprime (a@,c@)) Then
b@ = b@ + 1
Print " ";a@;
EndIf
Next
Print
Next
End
_kprime Param(2)
Local(2)
d@ = 0
For c@ = 2 Step 1 While (d@ < b@) * ((c@ * c@) < (a@ + 1))
Do While (a@ % c@) = 0
a@ = a@ / c@
d@ = d@ + 1
Loop
Next
Return (b@ = (d@ + (a@ > 1)))
For k = 1 To 5
Print "k = "; k; " : ";
i = 2
c = 0
Do While c < 10
If FUNC(_kPrime(i, k)) Then Print Using "__# "; i; : c = c + 1
i = i + 1
Loop
Print
Next
End
_kPrime
Param (2)
Local (2)
c@ = 0
For d@ = 2 To a@
Do While (a@ % d@) = 0
If c@ = b@ Then Unloop: Unloop: Return (0)
c@ = c@ + 1
a@ = a@ / d@
Loop
Next
Return (c@ = b@)
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176 0 OK, 0:200
Visual Basic .NET
Module Module1
Class KPrime
Public K As Integer
Public Function IsKPrime(number As Integer) As Boolean
Dim primes = 0
Dim p = 2
While p * p <= number AndAlso primes < K
While number Mod p = 0 AndAlso primes < K
number = number / p
primes = primes + 1
End While
p = p + 1
End While
If number > 1 Then
primes = primes + 1
End If
Return primes = K
End Function
Public Function GetFirstN(n As Integer) As List(Of Integer)
Dim result As New List(Of Integer)
Dim number = 2
While result.Count < n
If IsKPrime(number) Then
result.Add(number)
End If
number = number + 1
End While
Return result
End Function
End Class
Sub Main()
For Each k In Enumerable.Range(1, 5)
Dim kprime = New KPrime With {
.K = k
}
Console.WriteLine("k = {0}: {1}", k, String.Join(" ", kprime.GetFirstN(10)))
Next
End Sub
End Module
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
XBasic
' Almost prime
PROGRAM "almostprime"
VERSION "0.0002"
DECLARE FUNCTION Entry()
INTERNAL FUNCTION KPrime(n%%, k%%)
FUNCTION Entry()
FOR k@@ = 1 TO 5
PRINT "k ="; k@@; ":";
i%% = 2
c%% = 0
DO WHILE c%% < 10
IFT KPrime(i%%, k@@) THEN
PRINT FORMAT$(" ###", i%%);
INC c%%
END IF
INC i%%
LOOP
PRINT
NEXT k@@
END FUNCTION
FUNCTION KPrime(n%%, k%%)
f%% = 0
FOR i%% = 2 TO n%%
DO WHILE n%% MOD i%% = 0
IF f%% = k%% THEN RETURN $$FALSE
INC f%%
n%% = n%% \ i%%
LOOP
NEXT i%%
RETURN f%% = k%%
END FUNCTION
END PROGRAM
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Yabasic
// Returns boolean indicating whether n is k-almost prime
sub almostPrime(n, k)
local divisor, count
divisor = 2
while(count < (k + 1) and n <> 1)
if not mod(n, divisor) then
n = n / divisor
count = count + 1
else
divisor = divisor + 1
end if
wend
return count = k
end sub
// Generates table containing first ten k-almost primes for given k
sub kList(k, kTab())
local n, i
n = 2^k : i = 1
while(i < 11)
if almostPrime(n, k) then
kTab(i) = n
i = i + 1
end if
n = n + 1
wend
end sub
// Main procedure, displays results from five calls to kList()
dim kTab(10)
for k = 1 to 5
print "k = ", k, " : ";
kList(k, kTab())
for n = 1 to 10
print kTab(n), ", ";
next
print "..."
nextZX Spectrum Basic
10 FOR k=1 TO 5
20 PRINT k;":";
30 LET c=0: LET i=1
40 IF c=10 THEN GO TO 100
50 LET i=i+1
60 GO SUB 1000
70 IF r THEN PRINT " ";i;: LET c=c+1
90 GO TO 40
100 PRINT
110 NEXT k
120 STOP
1000 REM kprime
1010 LET p=2: LET n=i: LET f=0
1020 IF f=k OR (p*p)>n THEN GO TO 1100
1030 IF n/p=INT (n/p) THEN LET n=n/p: LET f=f+1: GO TO 1030
1040 LET p=p+1: GO TO 1020
1100 LET r=(f+(n>1)=k)
1110 RETURN
- Output:
1: 2 3 5 7 11 13 17 19 23 29 2: 4 6 9 10 14 15 21 22 25 26 3: 8 12 18 20 27 28 30 42 44 45 4: 16 24 36 40 54 56 60 81 84 88 5: 32 48 72 80 108 112 120 162 168 176
BCPL
get "libhdr"
let kprime(n, k) = valof
$( let f, p = 0, 2
while f<k & p*p<=n do
$( while n rem p = 0 do
$( n := n/p
f := f+1
$)
p := p+1
$)
if n > 1 then f := f + 1
resultis f = k
$)
let start() be
$( for k=1 to 5 do
$( let i, c = 2, 0
writef("k = %N:", k)
while c < 10 do
$( if kprime(i, k) then
$( writed(i, 4)
c := c+1
$)
i := i+1
$)
wrch('*N')
$)
$)- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Befunge
The extra spaces are to ensure it's readable on buggy interpreters that don't include a space after numeric output.
1>::48*"= k",,,,02p.":",01v
|^ v0!`\*:g40:<p402p300:+1<
K| >2g03g`*#v_ 1`03g+02g->|
F@>/03g1+03p>vpv+1\.:,*48 <
P#|!\g40%g40:<4>:9`>#v_\1^|
|^>#!1#`+#50#:^#+1,+5>#5$<|
- Output:
k = 1 : 2 3 5 7 11 13 17 19 23 29 k = 2 : 4 6 9 10 14 15 21 22 25 26 k = 3 : 8 12 18 20 27 28 30 42 44 45 k = 4 : 16 24 36 40 54 56 60 81 84 88 k = 5 : 32 48 72 80 108 112 120 162 168 176
C
#include <stdio.h>
int kprime(int n, int k)
{
int p, f = 0;
for (p = 2; f < k && p*p <= n; p++)
while (0 == n % p)
n /= p, f++;
return f + (n > 1) == k;
}
int main(void)
{
int i, c, k;
for (k = 1; k <= 5; k++) {
printf("k = %d:", k);
for (i = 2, c = 0; c < 10; i++)
if (kprime(i, k)) {
printf(" %d", i);
c++;
}
putchar('\n');
}
return 0;
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
C#
using System;
using System.Collections.Generic;
using System.Linq;
namespace AlmostPrime
{
class Program
{
static void Main(string[] args)
{
foreach (int k in Enumerable.Range(1, 5))
{
KPrime kprime = new KPrime() { K = k };
Console.WriteLine("k = {0}: {1}",
k, string.Join<int>(" ", kprime.GetFirstN(10)));
}
}
}
class KPrime
{
public int K { get; set; }
public bool IsKPrime(int number)
{
int primes = 0;
for (int p = 2; p * p <= number && primes < K; ++p)
{
while (number % p == 0 && primes < K)
{
number /= p;
++primes;
}
}
if (number > 1)
{
++primes;
}
return primes == K;
}
public List<int> GetFirstN(int n)
{
List<int> result = new List<int>();
for (int number = 2; result.Count < n; ++number)
{
if (IsKPrime(number))
{
result.Add(number);
}
}
return result;
}
}
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
C++
#include <cstdlib>
#include <iostream>
#include <sstream>
#include <iomanip>
#include <list>
bool k_prime(unsigned n, unsigned k) {
unsigned f = 0;
for (unsigned p = 2; f < k && p * p <= n; p++)
while (0 == n % p) { n /= p; f++; }
return f + (n > 1 ? 1 : 0) == k;
}
std::list<unsigned> primes(unsigned k, unsigned n) {
std::list<unsigned> list;
for (unsigned i = 2;list.size() < n;i++)
if (k_prime(i, k)) list.push_back(i);
return list;
}
int main(const int argc, const char* argv[]) {
using namespace std;
for (unsigned k = 1; k <= 5; k++) {
ostringstream os("");
const list<unsigned> l = primes(k, 10);
for (list<unsigned>::const_iterator i = l.begin(); i != l.end(); i++)
os << setw(4) << *i;
cout << "k = " << k << ':' << os.str() << endl;
}
return EXIT_SUCCESS;
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Clojure
(ns clojure.examples.almostprime
(:gen-class))
(defn divisors [n]
" Finds divisors by looping through integers 2, 3,...i.. up to sqrt (n) [note: rather than compute sqrt(), test with i*i <=n] "
(let [div (some #(if (= 0 (mod n %)) % nil) (take-while #(<= (* % %) n) (iterate inc 2)))]
(if div ; div = nil (if no divisor found else its the divisor)
(into [] (concat (divisors div) (divisors (/ n div)))) ; Concat the two divisors of the two divisors
[n]))) ; Number is prime so only itself as a divisor
(defn divisors-k [k n]
" Finds n numbers with k divisors. Does this by looping through integers 2, 3, ... filtering (passing) ones with k divisors and
taking the first n "
(->> (iterate inc 2) ; infinite sequence of numbers starting at 2
(map divisors) ; compute divisor of each element of sequence
(filter #(= (count %) k)) ; filter to take only elements with k divisors
(take n) ; take n elements from filtered sequence
(map #(apply * %)))) ; compute number by taking product of divisors
(println (for [k (range 1 6)]
(println "k:" k (divisors-k k 10))))
}
- Output:
(k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176) nil
CLU
kprime = proc (n,k: int) returns (bool)
f: int := 0
p: int := 2
while f<k & p*p<=n do
while n//p=0 do
n := n/p
f := f+1
end
p := p+1
end
if n>1 then f:=f+1 end
return(f=k)
end kprime
start_up = proc ()
po: stream := stream$primary_output()
for k: int in int$from_to(1,5) do
i: int := 2
c: int := 0
stream$puts(po, "k = " || int$unparse(k) || ":")
while c<10 do
if kprime(i,k) then
stream$putright(po, int$unparse(i), 4)
c := c+1
end
i := i+1
end
stream$putl(po, "")
end
end start_up- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
COBOL
IDENTIFICATION DIVISION.
PROGRAM-ID. ALMOST-PRIME.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 CONTROL-VARS.
03 K PIC 9.
03 I PIC 999.
03 SEEN PIC 99.
03 N PIC 999.
03 P PIC 99.
03 P-SQUARED PIC 9(4).
03 F PIC 99.
03 N-DIV-P PIC 999V999.
03 FILLER REDEFINES N-DIV-P.
05 NEXT-N PIC 999.
05 FILLER PIC 999.
88 N-DIVS-P VALUE ZERO.
01 OUT-VARS.
03 K-LN PIC X(70).
03 K-LN-PTR PIC 99.
03 LN-HDR.
05 FILLER PIC X(4) VALUE "K = ".
05 K-OUT PIC 9.
05 FILLER PIC X VALUE ":".
03 I-FMT.
05 FILLER PIC X VALUE SPACE.
05 I-OUT PIC ZZ9.
PROCEDURE DIVISION.
BEGIN.
PERFORM K-ALMOST-PRIMES VARYING K FROM 1 BY 1
UNTIL K IS GREATER THAN 5.
STOP RUN.
K-ALMOST-PRIMES.
MOVE SPACES TO K-LN.
MOVE 1 TO K-LN-PTR.
MOVE ZERO TO SEEN.
MOVE K TO K-OUT.
STRING LN-HDR DELIMITED BY SIZE INTO K-LN
WITH POINTER K-LN-PTR.
PERFORM I-K-ALMOST-PRIME VARYING I FROM 2 BY 1
UNTIL SEEN IS EQUAL TO 10.
DISPLAY K-LN.
I-K-ALMOST-PRIME.
MOVE ZERO TO F, P-SQUARED.
MOVE I TO N.
PERFORM PRIME-FACTOR VARYING P FROM 2 BY 1
UNTIL F IS NOT LESS THAN K
OR P-SQUARED IS GREATER THAN N.
IF N IS GREATER THAN 1, ADD 1 TO F.
IF F IS EQUAL TO K,
MOVE I TO I-OUT,
ADD 1 TO SEEN,
STRING I-FMT DELIMITED BY SIZE INTO K-LN
WITH POINTER K-LN-PTR.
PRIME-FACTOR.
MULTIPLY P BY P GIVING P-SQUARED.
DIVIDE N BY P GIVING N-DIV-P.
PERFORM DIVIDE-FACTOR UNTIL NOT N-DIVS-P.
DIVIDE-FACTOR.
MOVE NEXT-N TO N.
ADD 1 TO F.
DIVIDE N BY P GIVING N-DIV-P.
- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
Common Lisp
(defun start ()
(loop for k from 1 to 5
do (format t "k = ~a: ~a~%" k (collect-k-almost-prime k))))
(defun collect-k-almost-prime (k &optional (d 2) (lst nil))
(cond ((= (length lst) 10) (reverse lst))
((= (?-primality d) k) (collect-k-almost-prime k (+ d 1) (cons d lst)))
(t (collect-k-almost-prime k (+ d 1) lst))))
(defun ?-primality (n &optional (d 2) (c 0))
(cond ((> d (isqrt n)) (+ c 1))
((zerop (rem n d)) (?-primality (/ n d) d (+ c 1)))
(t (?-primality n (+ d 1) c))))
- Output:
k = 1: (2 3 5 7 11 13 17 19 23 29) k = 2: (4 6 9 10 14 15 21 22 25 26) k = 3: (8 12 18 20 27 28 30 42 44 45) k = 4: (16 24 36 40 54 56 60 81 84 88) k = 5: (32 48 72 80 108 112 120 162 168 176) NIL
Cowgol
include "cowgol.coh";
sub kprime(n: uint8, k: uint8): (kp: uint8) is
var p: uint8 := 2;
var f: uint8 := 0;
while f < k and p*p <= n loop
while 0 == n % p loop
n := n / p;
f := f + 1;
end loop;
p := p + 1;
end loop;
if n > 1 then
f := f + 1;
end if;
if f == k then
kp := 1;
else
kp := 0;
end if;
end sub;
var k: uint8 := 1;
while k <= 5 loop
print("k = ");
print_i8(k);
print(":");
var i: uint8 := 2;
var c: uint8 := 0;
while c < 10 loop
if kprime(i,k) != 0 then
print(" ");
print_i8(i);
c := c + 1;
end if;
i := i + 1;
end loop;
print_nl();
k := k + 1;
end loop;- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
D
This contains a copy of the function decompose from the Prime decomposition task.
import std.stdio, std.algorithm, std.traits;
Unqual!T[] decompose(T)(in T number) pure nothrow
in {
assert(number > 1);
} body {
typeof(return) result;
Unqual!T n = number;
for (Unqual!T i = 2; n % i == 0; n /= i)
result ~= i;
for (Unqual!T i = 3; n >= i * i; i += 2)
for (; n % i == 0; n /= i)
result ~= i;
if (n != 1)
result ~= n;
return result;
}
void main() {
enum outLength = 10; // 10 k-th almost primes.
foreach (immutable k; 1 .. 6) {
writef("K = %d: ", k);
auto n = 2; // The "current number" to be checked.
foreach (immutable i; 1 .. outLength + 1) {
while (n.decompose.length != k)
n++;
// Now n is K-th almost prime.
write(n, " ");
n++;
}
writeln;
}
}
- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
Delphi
program AlmostPrime;
{$APPTYPE CONSOLE}
function IsKPrime(const n, k: Integer): Boolean;
var
p, f, v: Integer;
begin
f := 0;
p := 2;
v := n;
while (f < k) and (p*p <= n) do begin
while (v mod p) = 0 do begin
v := v div p;
Inc(f);
end;
Inc(p);
end;
if v > 1 then Inc(f);
Result := f = k;
end;
var
i, c, k: Integer;
begin
for k := 1 to 5 do begin
Write('k = ', k, ':');
c := 0;
i := 2;
while c < 10 do begin
if IsKPrime(i, k) then begin
Write(' ', i);
Inc(c);
end;
Inc(i);
end;
WriteLn;
end;
end.
- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
Draco
proc nonrec kprime(word n, k) bool:
word f, p;
f := 0;
p := 2;
while f < k and p*p <= n do
while n%p = 0 do
n := n/p;
f := f+1
od;
p := p+1
od;
if n>1 then f+1 = k
else f = k
fi
corp
proc nonrec main() void:
byte k, i, c;
for k from 1 upto 5 do
write("k = ", k:1, ":");
i := 2;
c := 0;
while c < 10 do
if kprime(i,k) then
write(i:4);
c := c+1
fi;
i := i+1
od;
writeln()
od
corp- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
EasyLang
func kprime n k .
i = 2
while i <= n
while n mod i = 0
if f = k
return 0
.
f += 1
n /= i
.
i += 1
.
if f = k
return 1
.
return 0
.
for k = 1 to 5
write "k=" & k & " : "
i = 2
c = 0
while c < 10
if kprime i k = 1
write i & " "
c += 1
.
i += 1
.
print ""
.EchoLisp
Small numbers : filter the sequence [ 2 .. n]
(define (almost-prime? p k)
(= k (length (prime-factors p))))
(define (almost-primes k nmax)
(take (filter (rcurry almost-prime? k) [2 ..]) nmax))
(define (task (kmax 6) (nmax 10))
(for ((k [1 .. kmax]))
(write 'k= k '|)
(for-each write (almost-primes k nmax))
(writeln)))
- Output:
(task)
k= 1 | 2 3 5 7 11 13 17 19 23 29
k= 2 | 4 6 9 10 14 15 21 22 25 26
k= 3 | 8 12 18 20 27 28 30 42 44 45
k= 4 | 16 24 36 40 54 56 60 81 84 88
k= 5 | 32 48 72 80 108 112 120 162 168 176
Large numbers : generate - combinations with repetitions - k-almost-primes up to pmax.
(lib 'match)
(define-syntax-rule (: v i) (vector-ref v i))
(reader-infix ':) ;; abbrev (vector-ref v i) === [v : i]
(lib 'bigint)
(define cprimes (list->vector (primes 10000)))
;; generates next k-almost-prime < pmax
;; c = vector of k primes indices c[i] <= c[j]
;; p = vector of intermediate products prime[c[0]]*prime[c[1]]*..
;; p[k-1] is the generated k-almost-prime
;; increment one c[i] at each step
(define (almost-next pmax k c p)
(define almost-prime #f)
(define cp 0)
(for ((i (in-range (1- k) -1 -1))) ;; look backwards for c[i] to increment
(vector-set! c i (1+ [c : i])) ;; increment c[i]
(set! cp [cprimes : [c : i]])
(vector-set! p i (if (> i 0) (* [ p : (1- i)] cp) cp)) ;; update partial product
(when (< [p : i) pmax)
(set! almost-prime
(and ;; set followers to c[i] value
(for ((j (in-range (1+ i) k)))
(vector-set! c j [c : i])
(vector-set! p j (* [ p : (1- j)] cp))
#:break (>= [p : j] pmax) => #f )
[p : (1- k)]
) ;; // and
) ;; set!
) ;; when
#:break almost-prime
) ;; // for i
almost-prime )
;; not sorted list of k-almost-primes < pmax
(define (almost-primes k nmax)
(define base (expt 2 k)) ;; first one is 2^k
(define pmax (* base nmax))
(define c (make-vector k #0))
(define p (build-vector k (lambda(i) (expt #2 (1+ i)))))
(cons base
(for/list
((almost-prime (in-producer almost-next pmax k c p )))
almost-prime)))
- Output:
;; we want 500-almost-primes from the 10000-th.
(take (drop (list-sort < (almost-primes 500 10000)) 10000 ) 10)
(7241149198492252834202927258094752774597239286103014697435725917649659974371690699721153852986
440733637405206125678822081264723636566725108094369093648384
etc ...
;; The first one is 2^497 * 3 * 17 * 347 , same result as Haskell.
Elixir
defmodule Factors do
def factors(n), do: factors(n,2,[])
defp factors(1,_,acc), do: acc
defp factors(n,k,acc) when rem(n,k)==0, do: factors(div(n,k),k,[k|acc])
defp factors(n,k,acc) , do: factors(n,k+1,acc)
def kfactors(n,k), do: kfactors(n,k,1,1,[])
defp kfactors(_tn,tk,_n,k,_acc) when k == tk+1, do: IO.puts "done! "
defp kfactors(tn,tk,_n,k,acc) when length(acc) == tn do
IO.puts "K: #{k} #{inspect acc}"
kfactors(tn,tk,2,k+1,[])
end
defp kfactors(tn,tk,n,k,acc) do
case length(factors(n)) do
^k -> kfactors(tn,tk,n+1,k,acc++[n])
_ -> kfactors(tn,tk,n+1,k,acc)
end
end
end
Factors.kfactors(10,5)
- Output:
K: 1 [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] K: 2 [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] K: 3 [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] K: 4 [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] K: 5 [32, 48, 72, 80, 108, 112, 120, 162, 168, 176] done!
Erlang
Using the factors function from Prime_decomposition#Erlang.
-module(factors).
-export([factors/1,kfactors/0,kfactors/2]).
factors(N) ->
factors(N,2,[]).
factors(1,_,Acc) -> Acc;
factors(N,K,Acc) when N rem K == 0 ->
factors(N div K,K, [K|Acc]);
factors(N,K,Acc) ->
factors(N,K+1,Acc).
kfactors() -> kfactors(10,5,1,1,[]).
kfactors(N,K) -> kfactors(N,K,1,1,[]).
kfactors(_Tn,Tk,_N,K,_Acc) when K == Tk+1 -> io:fwrite("Done! ");
kfactors(Tn,Tk,N,K,Acc) when length(Acc) == Tn ->
io:format("K: ~w ~w ~n", [K, Acc]),
kfactors(Tn,Tk,2,K+1,[]);
kfactors(Tn,Tk,N,K,Acc) ->
case length(factors(N)) of K ->
kfactors(Tn,Tk, N+1,K, Acc ++ [ N ] );
_ ->
kfactors(Tn,Tk, N+1,K, Acc) end.
- Output:
9> factors:kfactors(10,5). K: 1 [2,3,5,7,11,13,17,19,23,29] K: 2 [4,6,9,10,14,15,21,22,25,26] K: 3 [8,12,18,20,27,28,30,42,44,45] K: 4 [16,24,36,40,54,56,60,81,84,88] K: 5 [32,48,72,80,108,112,120,162,168,176] Done! ok 10> factors:kfactors(15,10). K: 1 [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47] K: 2 [4,6,9,10,14,15,21,22,25,26,33,34,35,38,39] K: 3 [8,12,18,20,27,28,30,42,44,45,50,52,63,66,68] K: 4 [16,24,36,40,54,56,60,81,84,88,90,100,104,126,132] K: 5 [32,48,72,80,108,112,120,162,168,176,180,200,208,243,252] K: 6 [64,96,144,160,216,224,240,324,336,352,360,400,416,486,504] K: 7 [128,192,288,320,432,448,480,648,672,704,720,800,832,972,1008] K: 8 [256,384,576,640,864,896,960,1296,1344,1408,1440,1600,1664,1944,2016] K: 9 [512,768,1152,1280,1728,1792,1920,2592,2688,2816,2880,3200,3328,3888,4032] K: 10 [1024,1536,2304,2560,3456,3584,3840,5184,5376,5632,5760,6400,6656,7776,8064] Done! ok
ERRE
PROGRAM ALMOST_PRIME
!
! for rosettacode.org
!
!$INTEGER
PROCEDURE KPRIME(N,K->KP)
LOCAL P,F
FOR P=2 TO 999 DO
EXIT IF NOT((F<K) AND (P*P<=N))
WHILE (N MOD P)=0 DO
N/=P
F+=1
END WHILE
END FOR
KP=(F-(N>1)=K)
END PROCEDURE
BEGIN
PRINT(CHR$(12);) !CLS
FOR K=1 TO 5 DO
PRINT("k =";K;":";)
C=0
FOR I=2 TO 999 DO
EXIT IF NOT(C<10)
KPRIME(I,K->KP)
IF KP THEN
PRINT(I;)
C+=1
END IF
END FOR
PRINT
END FOR
END PROGRAM- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
F#
let rec genFactor (f, n) =
if f > n then None
elif n % f = 0 then Some (f, (f, n/f))
else genFactor (f+1, n)
let factorsOf (num) =
Seq.unfold (fun (f, n) -> genFactor (f, n)) (2, num)
let kFactors k = Seq.unfold (fun n ->
let rec loop m =
if Seq.length (factorsOf m) = k then m
else loop (m+1)
let next = loop n
Some(next, next+1)) 2
[1 .. 5]
|> List.iter (fun k ->
printfn "%A" (Seq.take 10 (kFactors k) |> Seq.toList))
- Output:
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29] [4; 6; 9; 10; 14; 15; 21; 22; 25; 26] [8; 12; 18; 20; 27; 28; 30; 42; 44; 45] [16; 24; 36; 40; 54; 56; 60; 81; 84; 88] [32; 48; 72; 80; 108; 112; 120; 162; 168; 176]
Factor
USING: formatting fry kernel lists lists.lazy locals
math.combinatorics math.primes.factors math.ranges sequences ;
IN: rosetta-code.almost-prime
: k-almost-prime? ( n k -- ? )
'[ factors _ <combinations> [ product ] map ]
[ [ = ] curry ] bi any? ;
:: first10 ( k -- seq )
10 0 lfrom [ k k-almost-prime? ] lfilter ltake list>array ;
5 [1,b] [ dup first10 "K = %d: %[%3d, %]\n" printf ] each
- Output:
K = 1: { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 }
K = 2: { 4, 6, 9, 10, 14, 15, 21, 22, 25, 26 }
K = 3: { 8, 12, 18, 20, 27, 28, 30, 42, 44, 45 }
K = 4: { 16, 24, 36, 40, 54, 56, 60, 81, 84, 88 }
K = 5: { 32, 48, 72, 80, 108, 112, 120, 162, 168, 176 }
FOCAL
01.10 F K=1,5;D 3
01.20 Q
02.10 S N=I;S P=1;S G=0
02.20 S P=P+1
02.30 I (K-G)2.7,2.7;I (N-P*P)2.7
02.40 S Z=FITR(N/P)
02.50 I (Z*P-N)2.2
02.60 S N=Z;S G=G+1;G 2.4
02.70 I (1-N)2.8;R
02.80 S G=G+1
03.10 T "K",%1,K,":"
03.20 S I=2;S C=0
03.30 D 2;I (G-K)3.6,3.4,3.6
03.40 T " ",%3,I
03.50 S C=C+1
03.60 S I=I+1
03.70 I (C-10)3.3
03.80 T !- Output:
K= 1: = 2 = 3 = 5 = 7 = 11 = 13 = 17 = 19 = 23 = 29 K= 2: = 4 = 6 = 9 = 10 = 14 = 15 = 21 = 22 = 25 = 26 K= 3: = 8 = 12 = 18 = 20 = 27 = 28 = 30 = 42 = 44 = 45 K= 4: = 16 = 24 = 36 = 40 = 54 = 56 = 60 = 81 = 84 = 88 K= 5: = 32 = 48 = 72 = 80 = 108 = 112 = 120 = 162 = 168 = 176
Fortran
program almost_prime
use iso_fortran_env, only: output_unit
implicit none
integer :: i, c, k
do k = 1, 5
write(output_unit,'(A3,x,I0,x,A1,x)', advance="no") "k =", k, ":"
i = 2
c = 0
do
if (c >= 10) exit
if (kprime(i, k)) then
write(output_unit,'(I0,x)', advance="no") i
c = c + 1
end if
i = i + 1
end do
write(output_unit,*)
end do
contains
pure function kprime(n, k)
integer, intent(in) :: n, k
logical :: kprime
integer :: p, f, i
kprime = .false.
f = 0
i = n
do p = 2, n
do
if (modulo(i, p) /= 0) exit
if (f == k) return
f = f + 1
i = i / p
end do
end do
kprime = f==k
end function kprime
end program almost_prime
- Output:
k = 1 : 2 3 5 7 11 13 17 19 23 29 k = 2 : 4 6 9 10 14 15 21 22 25 26 k = 3 : 8 12 18 20 27 28 30 42 44 45 k = 4 : 16 24 36 40 54 56 60 81 84 88 k = 5 : 32 48 72 80 108 112 120 162 168 176
Frink
for k = 1 to 5
{
n=2
count = 0
print["k=$k:"]
do
{
if length[factorFlat[n]] == k
{
print[" $n"]
count = count + 1
}
n = n + 1
} while count < 10
println[]
}Output:
k=1: 2 3 5 7 11 13 17 19 23 29 k=2: 4 6 9 10 14 15 21 22 25 26 k=3: 8 12 18 20 27 28 30 42 44 45 k=4: 16 24 36 40 54 56 60 81 84 88 k=5: 32 48 72 80 108 112 120 162 168 176
Futhark
let kprime(n: i32, k: i32): bool =
let (p,f) = (2, 0)
let (n,_,f) = loop (n, p, f) while f < k && p*p <= n do
let (n,f) = loop (n, f) while 0 == n % p do
(n/p, f+1)
in (n, p+1, f)
in f + (if n > 1 then 1 else 0) == k
let main(m: i32): [][]i32 =
let f k =
let ps = replicate 10 0
let (_,_,ps) = loop (i,c,ps) = (2,0,ps) while c < 10 do
if kprime(i,k) then
unsafe let ps[c] = i
in (i+1, c+1, ps)
else (i+1, c, ps)
in ps
in map f (1...m)
FutureBasic
local fn K_Prime( n as long, k as long ) as BOOL
long f = 0, i = 0
BOOL result
for i = 2 to n
while ( n mod i == 0 )
if f = k then exit fn = NO
f += 1
n /= i
wend
next
result = f = k
end fn = result
long i, c, k
for k = 1 to 5
printf @"k = %ld:\b", k
i = 2
c = 0
while ( c < 10 )
if ( fn K_Prime( i, k ) )
printf @"%4ld\b", i
c++
end if
i++
wend
print
next
HandleEvents- Output:
k = 1 : 2 3 5 7 11 13 17 19 23 29 k = 2 : 4 6 9 10 14 15 21 22 25 26 k = 3 : 8 12 18 20 27 28 30 42 44 45 k = 4 : 16 24 36 40 54 56 60 81 84 88 k = 5 : 32 48 72 80 108 112 120 162 168 176
Go
package main
import "fmt"
func kPrime(n, k int) bool {
nf := 0
for i := 2; i <= n; i++ {
for n%i == 0 {
if nf == k {
return false
}
nf++
n /= i
}
}
return nf == k
}
func gen(k, n int) []int {
r := make([]int, n)
n = 2
for i := range r {
for !kPrime(n, k) {
n++
}
r[i] = n
n++
}
return r
}
func main() {
for k := 1; k <= 5; k++ {
fmt.Println(k, gen(k, 10))
}
}
- Output:
1 [2 3 5 7 11 13 17 19 23 29] 2 [4 6 9 10 14 15 21 22 25 26] 3 [8 12 18 20 27 28 30 42 44 45] 4 [16 24 36 40 54 56 60 81 84 88] 5 [32 48 72 80 108 112 120 162 168 176]
Groovy
public class almostprime
{
public static boolean kprime(int n,int k)
{
int i,div=0;
for(i=2;(i*i <= n) && (div<k);i++)
{
while(n%i==0)
{
n = n/i;
div++;
}
}
return div + ((n > 1)?1:0) == k;
}
public static void main(String[] args)
{
int i,l,k;
for(k=1;k<=5;k++)
{
println("k = " + k + ":");
l = 0;
for(i=2;l<10;i++)
{
if(kprime(i,k))
{
print(i + " ");
l++;
}
}
println();
}
}
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Haskell
isPrime :: Integral a => a -> Bool
isPrime n = not $ any ((0 ==) . (mod n)) [2..(truncate $ sqrt $ fromIntegral n)]
primes :: [Integer]
primes = filter isPrime [2..]
isKPrime :: (Num a, Eq a) => a -> Integer -> Bool
isKPrime 1 n = isPrime n
isKPrime k n = any (isKPrime (k - 1)) sprimes
where
sprimes = map fst $ filter ((0 ==) . snd) $ map (divMod n) $ takeWhile (< n) primes
kPrimes :: (Num a, Eq a) => a -> [Integer]
kPrimes k = filter (isKPrime k) [2..]
main :: IO ()
main = flip mapM_ [1..5] $ \k ->
putStrLn $ "k = " ++ show k ++ ": " ++ (unwords $ map show (take 10 $ kPrimes k))
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Larger ks require more complicated methods:
primes = 2:3:[n | n <- [5,7..], foldr (\p r-> p*p > n || rem n p > 0 && r)
True (drop 1 primes)]
merge aa@(a:as) bb@(b:bs)
| a < b = a:merge as bb
| otherwise = b:merge aa bs
-- n-th item is all k-primes not divisible by any of the first n primes
notdivs k = f primes $ kprimes (k-1) where
f (p:ps) s = map (p*) s : f ps (filter ((/=0).(`mod`p)) s)
kprimes k
| k == 1 = primes
| otherwise = f (head ndk) (tail ndk) (tail $ map (^k) primes) where
ndk = notdivs k
-- tt is the thresholds for merging in next sequence
-- it is equal to "map head seqs", but don't do that
f aa@(a:as) seqs tt@(t:ts)
| a < t = a : f as seqs tt
| otherwise = f (merge aa $ head seqs) (tail seqs) ts
main = do
-- next line is for task requirement:
mapM_ (\x->print (x, take 10 $ kprimes x)) [1 .. 5]
putStrLn "\n10000th to 10100th 500-amost primes:"
mapM_ print $ take 100 $ drop 10000 $ kprimes 500
- Output:
(1,[2,3,5,7,11,13,17,19,23,29])
(2,[4,6,9,10,14,15,21,22,25,26])
(3,[8,12,18,20,27,28,30,42,44,45])
(4,[16,24,36,40,54,56,60,81,84,88])
(5,[32,48,72,80,108,112,120,162,168,176])
10000th to 10100th 500-amost primes:
7241149198492252834202927258094752774597239286103014697435725917649659974371690699721153852986440733637405206125678822081264723636566725108094369093648384
<...snipped 99 more equally unreadable numbers...>
Icon and Unicon
Works in both languages.
link "factors"
procedure main()
every writes(k := 1 to 5,": ") do
every writes(right(genKap(k),5)\10|"\n")
end
procedure genKap(k)
suspend (k = *factors(n := seq(q)), n)
end
Output:
->ap 1: 2 3 5 7 11 13 17 19 23 29 2: 4 6 9 10 14 15 21 22 25 26 3: 8 12 18 20 27 28 30 42 44 45 4: 16 24 36 40 54 56 60 81 84 88 5: 32 48 72 80 108 112 120 162 168 176 ->
Insitux
(function prime-sieve search siever sieved
(return-when (empty? siever) (.. vec sieved search))
(let [p ps] ((juxt 0 (skip 1)) siever))
(recur (remove #(div? % p) search)
(remove #(div? % p) ps)
(append p sieved)))
(function primes n
(prime-sieve (range 2 (inc n)) (range 2 (ceil (sqrt n))) []))
(function decompose n ps factors
(return-when (= n 1) factors)
(let div (find (div? n) ps))
(recur (/ n div) ps (append div factors)))
(function almost-prime up-to n k
(return-when (zero? up-to) [])
(let ps (primes n))
(if (= k (len (decompose n ps [])))
(prepend n (almost-prime (dec up-to) (inc n) k))
(almost-prime up-to (inc n) k)))
(function row n
(-> n
@(almost-prime 10 1)
(join " ")
@(str n (match n 1 "st" 2 "nd" 3 "rd" "th") " almost-primes: " )))
(join "\n" (map row (range 1 6)))- Output:
1st almost-primes: 2 3 5 7 11 13 17 19 23 29 2nd almost-primes: 4 6 9 10 14 15 21 22 25 26 3rd almost-primes: 8 12 18 20 27 28 30 42 44 45 4th almost-primes: 16 24 36 40 54 56 60 81 84 88 5th almost-primes: 32 48 72 80 108 112 120 162 168 176
J
(10 {. [:~.[:/:~[:,*/~)^:(i.5)~p:i.10
2 3 5 7 11 13 17 19 23 29
4 6 9 10 14 15 21 22 25 26
8 12 18 20 27 28 30 42 44 45
16 24 36 40 54 56 60 81 84 88
32 48 72 80 108 112 120 162 168 176
Explanation:
- Generate 10 primes.
- Multiply each of them by the first ten primes
- Sort and find unique values, take the first ten of those
- Multiply each of them by the first ten primes
- Sort and find unique values, take the first ten of those
- ...
The results of the odd steps in this procedure are the desired result.
Java
public class AlmostPrime {
public static void main(String[] args) {
for (int k = 1; k <= 5; k++) {
System.out.print("k = " + k + ":");
for (int i = 2, c = 0; c < 10; i++) {
if (kprime(i, k)) {
System.out.print(" " + i);
c++;
}
}
System.out.println("");
}
}
public static boolean kprime(int n, int k) {
int f = 0;
for (int p = 2; f < k && p * p <= n; p++) {
while (n % p == 0) {
n /= p;
f++;
}
}
return f + ((n > 1) ? 1 : 0) == k;
}
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
JavaScript
function almostPrime (n, k) {
var divisor = 2, count = 0
while(count < k + 1 && n != 1) {
if (n % divisor == 0) {
n = n / divisor
count = count + 1
} else {
divisor++
}
}
return count == k
}
for (var k = 1; k <= 5; k++) {
document.write("<br>k=", k, ": ")
var count = 0, n = 0
while (count <= 10) {
n++
if (almostPrime(n, k)) {
document.write(n, " ")
count++
}
}
}
- Output:
k=1: 2 3 5 7 11 13 17 19 23 29 31 k=2: 4 6 9 10 14 15 21 22 25 26 33 k=3: 8 12 18 20 27 28 30 42 44 45 50 k=4: 16 24 36 40 54 56 60 81 84 88 90 k=5: 32 48 72 80 108 112 120 162 168 176 180
jq
Infrastructure:
# Recent versions of jq (version > 1.4) have the following definition of "until":
def until(cond; next):
def _until:
if cond then . else (next|_until) end;
_until;
# relatively_prime(previous) tests whether the input integer is prime
# relative to the primes in the array "previous":
def relatively_prime(previous):
. as $in
| (previous|length) as $plen
# state: [found, ix]
| [false, 0]
| until( .[0] or .[1] >= $plen;
[ ($in % previous[.[1]]) == 0, .[1] + 1] )
| .[0] | not ;
# Emit a stream in increasing order of all primes (from 2 onwards)
# that are less than or equal to mx:
def primes(mx):
# The helper function, next, has arity 0 for tail recursion optimization;
# it expects its input to be the array of previously found primes:
def next:
. as $previous
| ($previous | .[length-1]) as $last
| if ($last >= mx) then empty
else ((2 + $last)
| until( relatively_prime($previous) ; . + 2)) as $nextp
| if $nextp <= mx
then $nextp, (( $previous + [$nextp] ) | next)
else empty
end
end;
if mx <= 1 then empty
elif mx == 2 then 2
else (2, 3, ( [2,3] | next))
end
;
# Return an array of the distinct prime factors of . in increasing order
def prime_factors:
# Return an array of prime factors of . given that "primes"
# is an array of relevant primes:
def pf(primes):
if . <= 1 then []
else . as $in
| if ($in | relatively_prime(primes)) then [$in]
else reduce primes[] as $p
([];
if ($in % $p) != 0 then .
else . + [$p] + (($in / $p) | pf(primes))
end)
end
| unique
end;
if . <= 1 then []
else . as $in
| pf( [ primes( (1+$in) | sqrt | floor) ] )
end;
# Return an array of prime factors of . repeated according to their multiplicities:
def prime_factors_with_multiplicities:
# Emit p according to the multiplicity of p
# in the input integer assuming p > 1
def multiplicity(p):
if . < p then empty
elif . == p then p
elif (. % p) == 0 then
((./p) | recurse( if (. % p) == 0 then (. / p) else empty end) | p)
else empty
end;
if . <= 1 then []
else . as $in
| prime_factors as $primes
| if ($in|relatively_prime($primes)) then [$in]
else reduce $primes[] as $p
([];
if ($in % $p) == 0 then . + [$in|multiplicity($p)] else . end )
end
end;isalmostprime
def isalmostprime(k): (prime_factors_with_multiplicities | length) == k;
# Emit a stream of the first N almost-k primes
def almostprimes(N; k):
if N <= 0 then empty
else
# state [remaining, candidate, answer]
[N, 1, null]
| recurse( if .[0] <= 0 then empty
elif (.[1] | isalmostprime(k)) then [.[0]-1, .[1]+1, .[1]]
else [.[0], .[1]+1, null]
end)
| .[2] | select(. != null)
end;The task:
range(1;6) as $k | "k=\($k): \([almostprimes(10;$k)])"- Output:
$ jq -c -r -n -f Almost_prime.jq
k=1: [2,3,5,7,11,13,17,19,23,29]
k=2: [4,6,9,10,14,15,21,22,25,26]
k=3: [8,12,18,20,27,28,30,42,44,45]
k=4: [16,24,36,40,54,56,60,81,84,88]
k=5: [32,48,72,80,108,112,120,162,168,176]
Julia
using Primes
isalmostprime(n::Integer, k::Integer) = sum(values(factor(n))) == k
function almostprimes(N::Integer, k::Integer) # return first N almost-k primes
P = Vector{typeof(k)}(undef,N)
i = 0; n = 2
while i < N
if isalmostprime(n, k) P[i += 1] = n end
n += 1
end
return P
end
for k in 1:5
println("$k-Almost-primes: ", join(almostprimes(10, k), ", "), "...")
end
- Output:
1-Almost-primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... 2-Almost-primes: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26... 3-Almost-primes: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45... 4-Almost-primes: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88... 5-Almost-primes: 32, 48, 72, 80, 108, 112, 120, 162, 168, 176...
Kotlin
fun Int.k_prime(x: Int): Boolean {
var n = x
var f = 0
var p = 2
while (f < this && p * p <= n) {
while (0 == n % p) { n /= p; f++ }
p++
}
return f + (if (n > 1) 1 else 0) == this
}
fun Int.primes(n : Int) : List<Int> {
var i = 2
var list = mutableListOf<Int>()
while (list.size < n) {
if (k_prime(i)) list.add(i)
i++
}
return list
}
fun main(args: Array<String>) {
for (k in 1..5)
println("k = $k: " + k.primes(10))
}
- Output:
k = 1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] k = 2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] k = 3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] k = 4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] k = 5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Lua
-- Returns boolean indicating whether n is k-almost prime
function almostPrime (n, k)
local divisor, count = 2, 0
while count < k + 1 and n ~= 1 do
if n % divisor == 0 then
n = n / divisor
count = count + 1
else
divisor = divisor + 1
end
end
return count == k
end
-- Generates table containing first ten k-almost primes for given k
function kList (k)
local n, kTab = 2^k, {}
while #kTab < 10 do
if almostPrime(n, k) then
table.insert(kTab, n)
end
n = n + 1
end
return kTab
end
-- Main procedure, displays results from five calls to kList()
for k = 1, 5 do
io.write("k=" .. k .. ": ")
for _, v in pairs(kList(k)) do
io.write(v .. ", ")
end
print("...")
end
- Output:
k=1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... k=2: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... k=3: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, ... k=4: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, ... k=5: 32, 48, 72, 80, 108, 112, 120, 162, 168, 176, ...
Maple
AlmostPrimes:=proc(k, numvalues::posint:=10)
local aprimes, i, intfactors;
aprimes := Array([]);
i := 0;
do
i := i + 1;
intfactors := ifactors(i)[2];
intfactors := [seq(seq(intfactors[i][1], j=1..intfactors[i][2]),i = 1..numelems(intfactors))];
if numelems(intfactors) = k then
ArrayTools:-Append(aprimes,i);
end if;
until numelems(aprimes) = 10:
aprimes;
end proc:
<seq( AlmostPrimes(i), i = 1..5 )>;- Output:
[[2, 3, 5, 7, 11, 13, 17, 19, 23, 29], [4, 6, 9, 10, 14, 15, 21, 22, 25, 26], [8, 12, 18, 20, 27, 28, 30, 42, 44, 45], [16, 24, 36, 40, 54, 56, 60, 81, 84, 88], [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]]
MAD
NORMAL MODE IS INTEGER
INTERNAL FUNCTION(NN,KK)
ENTRY TO KPRIME.
F = 0
N = NN
THROUGH SCAN, FOR P=2, 1, F.GE.KK .OR. P*P.G.N
DIV WHENEVER N.E.N/P*P
N = N/P
F = F+1
TRANSFER TO DIV
END OF CONDITIONAL
SCAN CONTINUE
WHENEVER N.G.1, F = F+1
FUNCTION RETURN F.E.KK
END OF FUNCTION
VECTOR VALUES KFMT = $5(S1,2HK=,I1,S1)*$
VECTOR VALUES PFMT = $5(I4,S1)*$
PRINT FORMAT KFMT, 1, 2, 3, 4, 5
DIMENSION KPR(50)
THROUGH FNDKPR, FOR K=1, 1, K.G.5
C=0
THROUGH FNDKPR, FOR I=2, 1, C.GE.10
WHENEVER KPRIME.(I,K)
KPR(C*5+K) = I
C = C+1
END OF CONDITIONAL
FNDKPR CONTINUE
THROUGH OUT, FOR C=0, 1, C.GE.10
OUT PRINT FORMAT PFMT, KPR(C*5+1), KPR(C*5+2), KPR(C*5+3),
0 KPR(C*5+4), KPR(C*5+5)
END OF PROGRAM- Output:
K=1 K=2 K=3 K=4 K=5 2 4 8 16 32 3 6 12 24 48 5 9 18 36 72 7 10 20 40 80 11 14 27 54 108 13 15 28 56 112 17 21 30 60 120 19 22 42 81 162 23 25 44 84 168 29 26 45 88 176
Mathematica / Wolfram Language
kprimes[k_,n_] :=
(* generates a list of the n smallest k-almost-primes *)
Module[{firstnprimes, runningkprimes = {}},
firstnprimes = Prime[Range[n]];
runningkprimes = firstnprimes;
Do[
runningkprimes =
Outer[Times, firstnprimes , runningkprimes ] // Flatten // Union // Take[#, n] & ;
(* only keep lowest n numbers in our running list *)
, {i, 1, k - 1}];
runningkprimes
]
(* now to create table with n=10 and k ranging from 1 to 5 *)
Table[Flatten[{"k = " <> ToString[i] <> ": ", kprimes[i, 10]}], {i,1,5}] // TableForm
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Maxima
/* Predicate function that checks k-almost primality for given integer n and parameter k */
k_almost_primep(n,k):=if integerp((n)^(1/k)) and primep((n)^(1/k)) then true else
lambda([x],(length(ifactors(x))=k and unique(map(second,ifactors(x)))=[1]) or (length(ifactors(x))<k and apply("+",map(second,ifactors(x)))=k))(n)$
/* Function that given a parameter k1 returns the first len k1-almost primes */
k_almost_prime_count(k1,len):=block(
count:len,
while length(sublist(makelist(i,i,count),lambda([x],k_almost_primep(x,k1))))<len do (count:count+1),
sublist(makelist(i,i,count),lambda([x],k_almost_primep(x,k1))))$
/* Test cases */
k_almost_prime_count(1,10);
k_almost_prime_count(2,10);
k_almost_prime_count(3,10);
k_almost_prime_count(4,10);
k_almost_prime_count(5,10);
- Output:
[2,3,5,7,11,13,17,19,23,29] [4,6,9,10,14,15,21,22,25,26] [8,12,18,20,27,28,30,42,44,45] [16,24,36,40,54,56,60,81,84,88] [32,48,72,80,108,112,120,162,168,176]
MiniScript
primeFactory = function(n=2)
if n < 2 then return ""
for i in range(2, n)
p = floor(n / i)
q = n % i
if not q then return str(i) + " " + str(primeFactory(p))
end for
return n
end function
getAlmostPrimes = function(k)
almost = []
n = 2
while almost.len < 10
primes = primeFactory(n).trim.split
if primes.len == k then almost.push(n)
n += 1
end while
return almost
end function
for i in range(1, 5)
print i + ": " + getAlmostPrimes(i)
end for
- Output:
]run 1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] 2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] 3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] 4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] 5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Modula-2
MODULE AlmostPrime;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE KPrime(n,k : INTEGER) : BOOLEAN;
VAR p,f : INTEGER;
BEGIN
f := 0;
p := 2;
WHILE (f<k) AND (p*p<=n) DO
WHILE n MOD p = 0 DO
n := n DIV p;
INC(f)
END;
INC(p)
END;
IF n>1 THEN
RETURN f+1 = k
END;
RETURN f = k
END KPrime;
VAR
buf : ARRAY[0..63] OF CHAR;
i,c,k : INTEGER;
BEGIN
FOR k:=1 TO 5 DO
FormatString("k = %i:", buf, k);
WriteString(buf);
i:=2;
c:=0;
WHILE c<10 DO
IF KPrime(i,k) THEN
FormatString(" %i", buf, i);
WriteString(buf);
INC(c)
END;
INC(i)
END;
WriteLn;
END;
ReadChar;
END AlmostPrime.
Nim
proc prime(k: int, listLen: int): seq[int] =
result = @[]
var
test: int = 2
curseur: int = 0
while curseur < listLen:
var
i: int = 2
compte = 0
n = test
while i <= n:
if (n mod i)==0:
n = n div i
compte += 1
else:
i += 1
if compte == k:
result.add(test)
curseur += 1
test += 1
for k in 1..5:
echo "k = ",k," : ",prime(k,10)
- Output:
k = 1 : @[2, 3, 5, 7, 11, 13, 17, 19, 23, 29] k = 2 : @[4, 6, 9, 10, 14, 15, 21, 22, 25, 26] k = 3 : @[8, 12, 18, 20, 27, 28, 30, 42, 44, 45] k = 4 : @[16, 24, 36, 40, 54, 56, 60, 81, 84, 88] k = 5 : @[32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Objeck
class Kth_Prime {
function : native : kPrime(n : Int, k : Int) ~ Bool {
f := 0;
for (p := 2; f < k & p*p <= n; p+=1;) {
while (0 = n % p) {
n /= p; f+=1;
};
};
return f + ((n > 1) ? 1 : 0) = k;
}
function : Main(args : String[]) ~ Nil {
for (k := 1; k <= 5; k+=1;) {
"k = {$k}:"->Print();
c := 0;
for (i := 2; c < 10; i+=1;) {
if (kPrime(i, k)) {
" {$i}"->Print();
c+=1;
};
};
'\n'->Print();
};
}
}- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Odin
package almostprime
import "core:fmt"
main :: proc() {
i, c, k: int
for k in 1 ..= 5 {
fmt.printf("k = %d:", k)
for i, c := 2, 0; c < 10; i += 1 {
if kprime(i, k) {
fmt.printf(" %v", i)
c += 1
}
}
fmt.printf("\n")
}
}
kprime :: proc(n: int, k: int) -> bool {
p, f: int = 0, 0
n := n
for p := 2; f < k && p * p <= n; p += 1 {
for (0 == n % p) {
n /= p
f += 1
}
}
return f + (n > 1 ? 1 : 0) == k
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Oforth
: kprime?( n k -- b )
| i |
0 2 n for: i [
while( n i /mod swap 0 = ) [ ->n 1+ ] drop
]
k ==
;
: table( k -- [] )
| l |
Array new dup ->l
2 while (l size 10 <>) [ dup k kprime? if dup l add then 1+ ]
drop
;- Output:
>#[ table .cr ] 5 each [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Onyx (wasm)
procedural
package main
use core {printf}
main :: () -> void {
printf("\n");
for k in 1..6 {
printf("k = {}:", k);
i := 2;
c: i32;
while c < 10 {
if kprime(i, k) {
printf(" {}", i);
c += 1;
}
i += 1;
}
printf("\n");
}
}
kprime :: (n: i32, k: i32) -> bool {
f: i32;
while p := 2; f < k && p * p <= n {
while n % p == 0 {
n /= p;
f += 1;
}
p += 1;
}
return f + (1 if n > 1 else 0) == k;
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
functional
//+optional-semicolons
use core {printf}
use core.iter
main :: () {
generator :=
iter.counter(1)
|> iter.map(k => .{
k = k, kprimes = kprime_iter(k)->take(10)
})
|> iter.take(5)
for val in generator {
printf("k = {}:", val.k)
for p in val.kprimes do printf(" {}", p)
printf("\n")
}
}
kprime_iter :: k =>
iter.counter(2)
|> iter.filter((i, [k]) => kprime(i, k))
kprime :: (n, k) => {
f := 0
for p in iter.counter(2) {
if f >= k do break
if p * p > n do break
while n % p == 0 {
n /= p
f += 1
}
}
return f + (1 if n > 1 else 0) == k
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
PARI/GP
almost(k)=my(n); for(i=1,10,while(bigomega(n++)!=k,); print1(n", "));
for(k=1,5,almost(k);print)- Output:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 32, 48, 72, 80, 108, 112, 120, 162, 168, 176,
Pascal
program AlmostPrime;
{$IFDEF FPC}
{$Mode Delphi}
{$ENDIF}
uses
primtrial;
var
i,K,cnt : longWord;
BEGIN
K := 1;
repeat
cnt := 0;
i := 2;
write('K=',K:2,':');
repeat
if isAlmostPrime(i,K) then
Begin
write(i:6,' ');
inc(cnt);
end;
inc(i);
until cnt = 9;
writeln;
inc(k);
until k > 10;
END.
- output
K= 1 : 2 3 5 7 11 13 17 19 23 29 K= 2 : 4 6 9 10 14 15 21 22 25 26 K= 3 : 8 12 18 20 27 28 30 42 44 45 K= 4 : 16 24 36 40 54 56 60 81 84 88 K= 5 : 32 48 72 80 108 112 120 162 168 176 K= 6 : 64 96 144 160 216 224 240 324 336 352 K= 7 : 128 192 288 320 432 448 480 648 672 704 K= 8 : 256 384 576 640 864 896 960 1296 1344 1408 K= 9 : 512 768 1152 1280 1728 1792 1920 2592 2688 2816 K=10 : 1024 1536 2304 2560 3456 3584 3840 5184 5376 5632
Perl
Using a CPAN module, which is simple and fast:
use ntheory qw/factor/;
sub almost {
my($k,$n) = @_;
my $i = 1;
map { $i++ while scalar factor($i) != $k; $i++ } 1..$n;
}
say "$_ : ", join(" ", almost($_,10)) for 1..5;
- Output:
1 : 2 3 5 7 11 13 17 19 23 29 2 : 4 6 9 10 14 15 21 22 25 26 3 : 8 12 18 20 27 28 30 42 44 45 4 : 16 24 36 40 54 56 60 81 84 88 5 : 32 48 72 80 108 112 120 162 168 176
or writing everything by hand:
use strict;
use warnings;
sub k_almost_prime;
for my $k ( 1 .. 5 ) {
my $almost = 0;
print join(", ", map {
1 until k_almost_prime ++$almost, $k;
"$almost";
} 1 .. 10), "\n";
}
sub nth_prime;
sub k_almost_prime {
my ($n, $k) = @_;
return if $n <= 1 or $k < 1;
my $which_prime = 0;
for my $count ( 1 .. $k ) {
while( $n % nth_prime $which_prime ) {
++$which_prime;
}
$n /= nth_prime $which_prime;
return if $n == 1 and $count != $k;
}
($n == 1) ? 1 : ();
}
BEGIN {
# This is loosely based on one of the python solutions
# to the RC Sieve of Eratosthenes task.
my @primes = (2, 3, 5, 7);
my $p_iter = 1;
my $p = $primes[$p_iter];
my $q = $p*$p;
my %sieve;
my $candidate = $primes[-1] + 2;
sub nth_prime {
my $n = shift;
return if $n < 0;
OUTER: while( $#primes < $n ) {
while( my $s = delete $sieve{$candidate} ) {
my $next = $s + $candidate;
$next += $s while exists $sieve{$next};
$sieve{$next} = $s;
$candidate += 2;
}
while( $candidate < $q ) {
push @primes, $candidate;
$candidate += 2;
next OUTER if exists $sieve{$candidate};
}
my $twop = 2 * $p;
my $next = $q + $twop;
$next += $twop while exists $sieve{$next};
$sieve{$next} = $twop;
$p = $primes[++$p_iter];
$q = $p * $p;
$candidate += 2;
}
return $primes[$n];
}
}
- Output:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 4, 6, 9, 10, 14, 15, 21, 22, 25, 26 8, 12, 18, 20, 27, 28, 30, 42, 44, 45 16, 24, 36, 40, 54, 56, 60, 81, 84, 88 32, 48, 72, 80, 108, 112, 120, 162, 168, 176
Phixmonti
/# Rosetta Code problem: http://rosettacode.org/wiki/Almost_prime
by Galileo, 06/2022 #/
include ..\Utilitys.pmt
def test tps over mod not enddef
def kprime?
>ps >ps
0 ( 2 tps ) for
test while
tps over / int ps> drop >ps
swap 1 + swap
test endwhile
drop
endfor
ps> drop
ps> ==
enddef
5 for >ps
2 ( )
len 10 < while over tps kprime? if over 0 put endif swap 1 + swap len 10 < endwhile
nip ps> drop
endfor
pstack- Output:
[[2, 3, 5, 7, 11, 13, 17, 19, 23, 29], [4, 6, 9, 10, 14, 15, 21, 22, 25, 26], [8, 12, 18, 20, 27, 28, 30, 42, 44, 45], [16, 24, 36, 40, 54, 56, 60, 81, 84, 88], [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]] === Press any key to exit ===
PHP
<?php
// Almost prime
function isKPrime($n, $k)
{
$f = 0;
for ($j = 2; $j <= $n; $j++) {
while ($n % $j == 0) {
if ($f == $k)
return false;
$f++;
$n = floor($n / $j);
} // while
} // for $j
return ($f == $k);
}
for ($k = 1; $k <= 5; $k++) {
echo "k = ", $k, ":";
$i = 2;
$c = 0;
while ($c < 10) {
if (isKPrime($i, $k)) {
echo " ", str_pad($i, 3, ' ', STR_PAD_LEFT);
$c++;
}
$i++;
}
echo PHP_EOL;
}
?>
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Picat
go =>
N = 10,
Ps = primes(100).take(N),
println(1=Ps),
T = Ps,
foreach(K in 2..5)
T := mul_take(Ps,T,N),
println(K=T)
end,
nl,
foreach(K in 6..25)
T := mul_take(Ps,T,N),
println(K=T)
end,
nl.
% take first N values of L1 x L2
mul_take(L1,L2,N) = [I*J : I in L1, J in L2, I<=J].sort_remove_dups().take(N).
take(L,N) = [L[I] : I in 1..N].- Output:
1 = [2,3,5,7,11,13,17,19,23,29] 2 = [4,6,9,10,14,15,21,22,25,26] 3 = [8,12,18,20,27,28,30,42,44,45] 4 = [16,24,36,40,54,56,60,81,84,88] 5 = [32,48,72,80,108,112,120,162,168,176] 6 = [64,96,144,160,216,224,240,324,336,352] 7 = [128,192,288,320,432,448,480,648,672,704] 8 = [256,384,576,640,864,896,960,1296,1344,1408] 9 = [512,768,1152,1280,1728,1792,1920,2592,2688,2816] 10 = [1024,1536,2304,2560,3456,3584,3840,5184,5376,5632] 11 = [2048,3072,4608,5120,6912,7168,7680,10368,10752,11264] 12 = [4096,6144,9216,10240,13824,14336,15360,20736,21504,22528] 13 = [8192,12288,18432,20480,27648,28672,30720,41472,43008,45056] 14 = [16384,24576,36864,40960,55296,57344,61440,82944,86016,90112] 15 = [32768,49152,73728,81920,110592,114688,122880,165888,172032,180224] 16 = [65536,98304,147456,163840,221184,229376,245760,331776,344064,360448] 17 = [131072,196608,294912,327680,442368,458752,491520,663552,688128,720896] 18 = [262144,393216,589824,655360,884736,917504,983040,1327104,1376256,1441792] 19 = [524288,786432,1179648,1310720,1769472,1835008,1966080,2654208,2752512,2883584] 20 = [1048576,1572864,2359296,2621440,3538944,3670016,3932160,5308416,5505024,5767168] 21 = [2097152,3145728,4718592,5242880,7077888,7340032,7864320,10616832,11010048,11534336] 22 = [4194304,6291456,9437184,10485760,14155776,14680064,15728640,21233664,22020096,23068672] 23 = [8388608,12582912,18874368,20971520,28311552,29360128,31457280,42467328,44040192,46137344] 24 = [16777216,25165824,37748736,41943040,56623104,58720256,62914560,84934656,88080384,92274688] 25 = [33554432,50331648,75497472,83886080,113246208,117440512,125829120,169869312,176160768,184549376]
PL/I
almost_prime: procedure options(main);
kprime: procedure(nn, k) returns(bit);
declare (n, nn, k, p, f) fixed;
f = 0;
n = nn;
do p=2 repeat(p+1) while(f<k & p*p <= n);
do n=n repeat(n/p) while(mod(n,p) = 0);
f = f+1;
end;
end;
return(f + (n>1) = k);
end kprime;
declare (i, c, k) fixed;
do k=1 to 5;
put edit('k = ',k,':') (A,F(1),A);
c = 0;
do i=2 repeat(i+1) while(c<10);
if kprime(i,k) then do;
put edit(i) (F(4));
c = c+1;
end;
end;
put skip;
end;
end almost_prime;- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
PL/M
100H:
BDOS: PROCEDURE (FN, ARG); DECLARE FN BYTE, ARG ADDRESS; GO TO 5; END BDOS;
EXIT: PROCEDURE; CALL BDOS(0,0); END EXIT;
PRINT: PROCEDURE (S); DECLARE S ADDRESS; CALL BDOS(9,S); END PRINT;
PRINT$NUMBER: PROCEDURE (N);
DECLARE S (4) BYTE INITIAL ('...$');
DECLARE P ADDRESS, (N, C BASED P) BYTE;
P = .S(3);
DIGIT:
P = P - 1;
C = N MOD 10 + '0';
N = N / 10;
IF N > 0 THEN GO TO DIGIT;
CALL PRINT(P);
END PRINT$NUMBER;
KPRIME: PROCEDURE (N, K) BYTE;
DECLARE (N, K, P, F) BYTE;
F = 0;
P = 2;
DO WHILE F < K AND P*P <= N;
DO WHILE N MOD P = 0;
N = N/P;
F = F+1;
END;
P = P+1;
END;
IF N > 1 THEN F = F + 1;
RETURN F = K;
END KPRIME;
DECLARE (I, C, K) BYTE;
DO K=1 TO 5;
CALL PRINT(.'K = $');
CALL PRINT$NUMBER(K);
CALL PRINT(.':$');
C = 0;
I = 2;
DO WHILE C < 10;
IF KPRIME(I, K) THEN DO;
CALL PRINT(.' $');
CALL PRINT$NUMBER(I);
C = C+1;
END;
I = I+1;
END;
CALL PRINT(.(13,10,'$'));
END;
CALL EXIT;
EOF- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
Phix
sequence res = columnize({tagset(5)}) -- ie {{1},{2},{3},{4},{5}} integer n = 2, found = 0 while found<50 do integer l = length(prime_factors(n,true)) if l<=5 and length(res[l])<=10 then res[l] &= n found += 1 end if n += 1 end while string fmt = "k = %d: "&join(repeat("%4d",10))&"\n" for i=1 to 5 do printf(1,fmt,res[i]) end for
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
PicoLisp
(de factor (N)
(make
(let
(D 2
L (1 2 2 . (4 2 4 2 4 6 2 6 .))
M (sqrt N) )
(while (>= M D)
(if (=0 (% N D))
(setq M
(sqrt (setq N (/ N (link D)))) )
(inc 'D (pop 'L)) ) )
(link N) ) ) )
(de almost (N)
(let (X 2 Y 0)
(make
(loop
(when (and (nth (factor X) N) (not (cdr @)))
(link X)
(inc 'Y) )
(T (= 10 Y) 'done)
(inc 'X) ) ) ) )
(for I 5
(println I '-> (almost I) ) )
(bye)Potion
# Converted from C
kprime = (n, k):
p = 2, f = 0
while (f < k && p*p <= n):
while (0 == n % p):
n /= p
f++.
p++.
n = if (n > 1): 1.
else: 0.
f + n == k.
1 to 5 (k):
"k = " print, k print, ":" print
i = 2, c = 0
while (c < 10):
if (kprime(i, k)): " " print, i print, c++.
i++
.
"" say.C and Potion take 0.006s, Perl5 0.028s
Prolog
% almostPrime(K, +Take, List) succeeds if List can be unified with the
% first Take K-almost-primes.
% Notice that K need not be specified.
% To avoid having to cache or recompute the first Take primes, we define
% almostPrime/3 in terms of almostPrime/4 as follows:
%
almostPrime(K, Take, List) :-
% Compute the list of the first Take primes:
nPrimes(Take, Primes),
almostPrime(K, Take, Primes, List).
almostPrime(1, Take, Primes, Primes).
almostPrime(K, Take, Primes, List) :-
generate(2, K), % generate K >= 2
K1 is K - 1,
almostPrime(K1, Take, Primes, L),
multiplylist( Primes, L, Long),
sort(Long, Sorted), % uniquifies
take(Take, Sorted, List).
That's it. The rest is machinery. For portability, a compatibility section is included below.
nPrimes( M, Primes) :- nPrimes( [2], M, Primes).
nPrimes( Accumulator, I, Primes) :-
next_prime(Accumulator, Prime),
append(Accumulator, [Prime], Next),
length(Next, N),
( N = I -> Primes = Next; nPrimes( Next, I, Primes)).
% next_prime(+Primes, NextPrime) succeeds if NextPrime is the next
% prime after a list, Primes, of consecutive primes starting at 2.
next_prime([2], 3).
next_prime([2|Primes], P) :-
last(Primes, PP),
P2 is PP + 2,
generate(P2, N),
1 is N mod 2, % odd
Max is floor(sqrt(N+1)), % round-off paranoia
forall( (member(Prime, [2|Primes]),
(Prime =< Max -> true
; (!, fail))), N mod Prime > 0 ),
!,
P = N.
% multiply( +A, +List, Answer )
multiply( A, [], [] ).
multiply( A, [X|Xs], [AX|As] ) :-
AX is A * X,
multiply(A, Xs, As).
% multiplylist( L1, L2, List ) succeeds if List is the concatenation of X * L2
% for successive elements X of L1.
multiplylist( [], B, [] ).
multiplylist( [A|As], B, List ) :-
multiply(A, B, L1),
multiplylist(As, B, L2),
append(L1, L2, List).
take(N, List, Head) :-
length(Head, N),
append(Head,X,List).
%%%%% compatibility section %%%%%
:- if(current_prolog_flag(dialect, yap)).
generate(Min, I) :- between(Min, inf, I).
append([],L,L).
append([X|Xs], L, [X|Ls]) :- append(Xs,L,Ls).
:- endif.
:- if(current_prolog_flag(dialect, swi)).
generate(Min, I) :- between(Min, inf, I).
:- endif.
:- if(current_prolog_flag(dialect, yap)).
append([],L,L).
append([X|Xs], L, [X|Ls]) :- append(Xs,L,Ls).
last([X], X).
last([_|Xs],X) :- last(Xs,X).
:- endif.
:- if(current_prolog_flag(dialect, gprolog)).
generate(Min, I) :-
current_prolog_flag(max_integer, Max),
between(Min, Max, I).
:- endif.
Example using SWI-Prolog:
?- between(1,5,I), (almostPrime(I, 10, L) -> writeln(L)), fail. [2,3,5,7,11,13,17,19,23,29] [4,6,9,10,14,15,21,22,25,26] [8,12,18,20,27,28,30,42,44,45] [16,24,36,40,54,56,60,81,84,88] [32,48,72,80,108,112,120,162,168,176] ?- time( (almostPrime(5, 10, L), writeln(L))). [32,48,72,80,108,112,120,162,168,176] % 1,906 inferences, 0.001 CPU in 0.001 seconds (84% CPU, 2388471 Lips)
Processing
void setup() {
for (int i = 1; i <= 5; i++) {
int count = 0;
print("k = " + i + ": ");
int n = 2;
while (count < 10) {
if (isAlmostPrime(i, n)) {
count++;
print(n + " ");
}
n++;
}
println();
}
}
boolean isAlmostPrime(int k, int n) {
if (countPrimeFactors(n) == k) {
return true;
} else {
return false;
}
}
int countPrimeFactors(int n) {
int count = 0;
int i = 2;
while (n > 1) {
if (n % i == 0) {
n /= i;
count++;
} else {
i++;
}
}
return count;
}- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Python
This imports Prime decomposition#Python
from prime_decomposition import decompose
from itertools import islice, count
try:
from functools import reduce
except:
pass
def almostprime(n, k=2):
d = decompose(n)
try:
terms = [next(d) for i in range(k)]
return reduce(int.__mul__, terms, 1) == n
except:
return False
if __name__ == '__main__':
for k in range(1,6):
print('%i: %r' % (k, list(islice((n for n in count() if almostprime(n, k)), 10))))
- Output:
1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] 2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] 3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] 4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] 5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
An updated version with no import dependencies.
# k-Almost-primes
# Python 3.6.3
# no imports
# author: manuelcaeiro | https://github.com/manuelcaeiro
def prime_factors(m=2):
for i in range(2, m):
r, q = divmod(m, i)
if not q:
return [i] + prime_factors(r)
return [m]
def k_almost_primes(n, k=2):
multiples = set()
lists = list()
for x in range(k+1):
lists.append([])
for i in range(2, n+1):
if i not in multiples:
if len(lists[1]) < 10:
lists[1].append(i)
multiples.update(range(i*i, n+1, i))
print("k=1: {}".format(lists[1]))
for j in range(2, k+1):
for m in multiples:
l = prime_factors(m)
ll = len(l)
if ll == j and len(lists[j]) < 10:
lists[j].append(m)
print("k={}: {}".format(j, lists[j]))
k_almost_primes(200, 5)
# try:
#k_almost_primes(6000, 10)
- Output:
>>> %Run k_almost_primes.py k=1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] k=2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] k=3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] k=4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] k=5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Quackery
primefactors is defined at Prime decomposition#Quackery.
[ stack ] is quantity ( --> s )
[ stack ] is factors ( --> s )
[ factors put
quantity put
[] 1
[ over size
quantity share != while
1+ dup primefactors
size factors share = if
[ tuck join swap ]
again ]
drop
factors release
quantity release ] is almostprimes ( n n --> [ )
5 times
[ 10 i^ 1+ dup echo sp
almostprimes echo cr ]- Output:
1 [ 2 3 5 7 11 13 17 19 23 29 ] 2 [ 4 6 9 10 14 15 21 22 25 26 ] 3 [ 8 12 18 20 27 28 30 42 44 45 ] 4 [ 16 24 36 40 54 56 60 81 84 88 ] 5 [ 32 48 72 80 108 112 120 162 168 176 ]
R
This uses the function from Prime decomposition#R
#===============================================================
# Find k-Almost-primes
# R implementation
#===============================================================
#---------------------------------------------------------------
# Function for prime factorization from Rosetta Code
#---------------------------------------------------------------
findfactors <- function(n) {
d <- c()
div <- 2; nxt <- 3; rest <- n
while( rest != 1 ) {
while( rest%%div == 0 ) {
d <- c(d, div)
rest <- floor(rest / div)
}
div <- nxt
nxt <- nxt + 2
}
d
}
#---------------------------------------------------------------
# Find k-Almost-primes
#---------------------------------------------------------------
almost_primes <- function(n = 10, k = 5) {
# Set up matrix for storing of the results
res <- matrix(NA, nrow = k, ncol = n)
rownames(res) <- paste("k = ", 1:k, sep = "")
colnames(res) <- rep("", n)
# Loop over k
for (i in 1:k) {
tmp <- 1
while (any(is.na(res[i, ]))) { # Keep looping if there are still missing entries in the result-matrix
if (length(findfactors(tmp)) == i) { # Check number of factors
res[i, which.max(is.na(res[i, ]))] <- tmp
}
tmp <- tmp + 1
}
}
print(res)
}
- Output:
k = 1 2 3 5 7 11 13 17 19 23 29 k = 2 4 6 9 10 14 15 21 22 25 26 k = 3 8 12 18 20 27 28 30 42 44 45 k = 4 16 24 36 40 54 56 60 81 84 88 k = 5 32 48 72 80 108 112 120 162 168 176
Racket
#lang racket
(require (only-in math/number-theory factorize))
(define ((k-almost-prime? k) n)
(= k (for/sum ((f (factorize n))) (cadr f))))
(define KAP-table-values
(for/list ((k (in-range 1 (add1 5))))
(define kap? (k-almost-prime? k))
(for/list ((j (in-range 10)) (i (sequence-filter kap? (in-naturals 1))))
i)))
(define (format-table t)
(define longest-number-length
(add1 (order-of-magnitude (argmax order-of-magnitude (cons (length t) (apply append t))))))
(define (fmt-val v) (~a v #:width longest-number-length #:align 'right))
(string-join
(for/list ((r t) (k (in-naturals 1)))
(string-append
(format "║ k = ~a║ " (fmt-val k))
(string-join (for/list ((c r)) (fmt-val c)) "| ")
"║"))
"\n"))
(displayln (format-table KAP-table-values))
- Output:
║ k = 1║ 2| 3| 5| 7| 11| 13| 17| 19| 23| 29║ ║ k = 2║ 4| 6| 9| 10| 14| 15| 21| 22| 25| 26║ ║ k = 3║ 8| 12| 18| 20| 27| 28| 30| 42| 44| 45║ ║ k = 4║ 16| 24| 36| 40| 54| 56| 60| 81| 84| 88║ ║ k = 5║ 32| 48| 72| 80| 108| 112| 120| 162| 168| 176║
Raku
(formerly Perl 6)
sub is-k-almost-prime($n is copy, $k) returns Bool {
loop (my ($p, $f) = 2, 0; $f < $k && $p*$p <= $n; $p++) {
$n /= $p, $f++ while $n %% $p;
}
$f + ($n > 1) == $k;
}
for 1 .. 5 -> $k {
say ~.[^10]
given grep { is-k-almost-prime($_, $k) }, 2 .. *
}
- Output:
2 3 5 7 11 13 17 19 23 29 4 6 9 10 14 15 21 22 25 26 8 12 18 20 27 28 30 42 44 45 16 24 36 40 54 56 60 81 84 88 32 48 72 80 108 112 120 162 168 176
Here is a solution with identical output based on the factors routine from Count_in_factors#Raku (to be included manually until we decide where in the distribution to put it).
constant @primes = 2, |(3, 5, 7 ... *).grep: *.is-prime;
multi sub factors(1) { 1 }
multi sub factors(Int $remainder is copy) {
gather for @primes -> $factor {
# if remainder < factor², we're done
if $factor * $factor > $remainder {
take $remainder if $remainder > 1;
last;
}
# How many times can we divide by this prime?
while $remainder %% $factor {
take $factor;
last if ($remainder div= $factor) === 1;
}
}
}
constant @factory = lazy 0..* Z=> flat (0, 0, map { +factors($_) }, 2..*);
sub almost($n) { map *.key, grep *.value == $n, @factory }
put almost($_)[^10] for 1..5;
REXX
Version 1 naive solution
The method used is to count the number of factors in the number to determine the K-primality.
The first three k-almost primes for each K group are computed directly (rather than found).
/*REXX program computes and displays the first N K─almost primes from 1 ──► K. */
parse arg N K . /*get optional arguments from the C.L. */
if N=='' | N=="," then N=10 /*N not specified? Then use default.*/
if K=='' | K=="," then K= 5 /*K " " " " " */
/*W: is the width of K, used for output*/
do m=1 for K; $=2**m; fir=$ /*generate & assign 1st K─almost prime.*/
#=1; if #==N then leave /*#: K─almost primes; Enough are found?*/
#=2; $=$ 3*(2**(m-1)) /*generate & append 2nd K─almost prime.*/
if #==N then leave /*#: K─almost primes; Enough are found?*/
if m==1 then _=fir + fir /* [↓] gen & append 3rd K─almost prime*/
else do; _=9 * (2**(m-2)); #=3; $=$ _; end
do j=_ + m - 1 until #==N /*process an K─almost prime N times.*/
if factr()\==m then iterate /*not the correct K─almost prime? */
#=# + 1; $=$ j /*bump K─almost counter; append it to $*/
end /*j*/ /* [↑] generate N K─almost primes.*/
say right(m, length(K))"─almost ("N') primes:' $
end /*m*/ /* [↑] display a line for each K─prime*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
factr: z=j; do f=0 while z// 2==0; z=z% 2; end /*divisible by 2.*/
do f=f while z// 3==0; z=z% 3; end /*divisible " 3.*/
do f=f while z// 5==0; z=z% 5; end /*divisible " 5.*/
do f=f while z// 7==0; z=z% 7; end /*divisible " 7.*/
do f=f while z//11==0; z=z%11; end /*divisible " 11.*/
do f=f while z//13==0; z=z%13; end /*divisible " 13.*/
do p=17 by 6 while p<=z /*insure P isn't divisible by three. */
parse var p '' -1 _ /*obtain the right─most decimal digit. */
/* [↓] fast check for divisible by 5. */
if _\==5 then do; do f=f+1 while z//p==0; z=z%p; end; f=f-1; end /*÷ by P? */
if _ ==3 then iterate /*fast check for X divisible by five.*/
x=p+2; do f=f+1 while z//x==0; z=z%x; end; f=f-1 /*÷ by X? */
end /*i*/ /* [↑] find all the factors in Z. */
if f==0 then return 1 /*if prime (f==0), then return unity.*/
return f /*return to invoker the number of divs.*/
- output when using the default input:
1─almost (10) primes: 2 3 5 7 11 13 17 19 23 29 2─almost (10) primes: 4 6 9 10 14 15 21 22 25 26 3─almost (10) primes: 8 12 18 20 27 28 30 42 44 45 4─almost (10) primes: 16 24 36 40 54 56 60 81 84 88 5─almost (10) primes: 32 48 72 80 108 112 120 162 168 176 0.006 seconds (Regina)
- output when using the input of: 20 12
1─almost (20) primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 2─almost (20) primes: 4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 3─almost (20) primes: 8 12 18 20 27 28 30 42 44 45 50 52 63 66 68 70 75 76 78 92 4─almost (20) primes: 16 24 36 40 54 56 60 81 84 88 90 100 104 126 132 135 136 140 150 152 5─almost (20) primes: 32 48 72 80 108 112 120 162 168 176 180 200 208 243 252 264 270 272 280 300 6─almost (20) primes: 64 96 144 160 216 224 240 324 336 352 360 400 416 486 504 528 540 544 560 600 7─almost (20) primes: 128 192 288 320 432 448 480 648 672 704 720 800 832 972 1008 1056 1080 1088 1120 1200 8─almost (20) primes: 256 384 576 640 864 896 960 1296 1344 1408 1440 1600 1664 1944 2016 2112 2160 2176 2240 2400 9─almost (20) primes: 512 768 1152 1280 1728 1792 1920 2592 2688 2816 2880 3200 3328 3888 4032 4224 4320 4352 4480 4800 10─almost (20) primes: 1024 1536 2304 2560 3456 3584 3840 5184 5376 5632 5760 6400 6656 7776 8064 8448 8640 8704 8960 9600 11─almost (20) primes: 2048 3072 4608 5120 6912 7168 7680 10368 10752 11264 11520 12800 13312 15552 16128 16896 17280 17408 17920 19200 12─almost (20) primes: 4096 6144 9216 10240 13824 14336 15360 20736 21504 22528 23040 25600 26624 31104 32256 33792 34560 34816 35840 38400 22.380 seconds (Regina)
Version 2: optimized
This optimized REXX version can be over a hundred times faster than the naive version.
Some of the optimizations are:
- calculating the first 2(K-1) K─almost primes for each K group
- generating the primes (up to the limit) instead of dividing by (most) divisors.
- extending the up-front prime divisors in the factr function.
The 1st optimization (bullet) allows the direct computation (instead of searching) of all K─almost primes up to the first odd prime in the list.
Once the required primes are generated, the finding of the K─almost primes is almost instantaneous.
/*REXX program computes and displays the first N K─almost primes from 1 ──► K. */
parse arg N K . /*obtain optional arguments from the CL*/
if N=='' | N==',' then N=10 /*N not specified? Then use default.*/
if K=='' | K==',' then K= 5 /*K " " " " " */
nn=N; N=abs(N); w=length(K) /*N positive? Then show K─almost primes*/
limit= (2**K) * N / 2 /*this is the limit for most K-primes. */
if N==1 then limit=limit * 2 /* " " " " " a N of 1.*/
if K==1 then limit=limit * 4 /* " " " " " a K─prime " 2.*/
if K==2 then limit=limit * 2 /* " " " " " " " " 4.*/
if K==3 then limit=limit * 3 % 2 /* " " " " " " " " 8.*/
call genPrimes limit + 1 /*generate primes up to the LIMIT + 1.*/
say 'The highest prime computed: ' @.# " (under the limit of " limit').'
say /* [↓] define where 1st K─prime is odd*/
d.=0; d.2= 2; d.3 = 4; d.4 = 7; d.5 = 13; d.6 = 22; d.7 = 38; d.8=63
d.9=102; d.10=168; d.11=268; d.12=426; d.13=673; d.14=1064
d!=0
do m=1 for K; d!=max(d!,d.m) /*generate & assign 1st K─almost prime.*/
mr=right(m,w); mm=m-1
$=; do #=1 to min(N, d!) /*assign some doubled K─almost primes. */
$=$ d.mm.# * 2
end /*#*/
#=#-1
if m==1 then from=2
else from=1 + word($, words($) )
do j=from until #==N /*process an K─almost prime N times.*/
if factr()\==m then iterate /*not the correct K─almost prime? */
#=#+1; $=$ j /*bump K─almost counter; append it to $*/
end /*j*/ /* [↑] generate N K─almost primes.*/
if nn>0 then say mr"─almost ("N') primes:' $
else say ' the last' mr "K─almost prime: " word($, words($))
/* [↓] assign K─almost primes.*/
do q=1 for #; d.m.q=word($,q) ; end /*q*/
do q=1 for #; if d.m.q\==d.mm.q*2 then leave; end /*q*/
/* [↑] count doubly-duplicates*/
/*──── say copies('─',40) 'for ' m", " q-1 'numbers were doubly─duplicated.' ────*/
/*──── say ────*/
end /*m*/ /* [↑] display a line for each K─prime*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
factr: if #.j\==. then return #.j
z=j; do f=0 while z// 2==0; z=z% 2; end /*÷ by 2*/
do f=f while z// 3==0; z=z% 3; end /*÷ " 3*/
do f=f while z// 5==0; z=z% 5; end /*÷ " 5*/
do f=f while z// 7==0; z=z% 7; end /*÷ " 7*/
do f=f while z//11==0; z=z%11; end /*÷ " 11*/
do f=f while z//13==0; z=z%13; end /*÷ " 13*/
do f=f while z//17==0; z=z%17; end /*÷ " 17*/
do f=f while z//19==0; z=z%19; end /*÷ " 19*/
do i=9 while @.i<=z; d=@.i /*divide by some higher primes. */
do f=f while z//d==0; z=z%d; end /*is Z divisible by the prime D ? */
end /*i*/ /* [↑] find all factors in Z. */
if f==0 then f=1; #.j=f; return f /*Is prime (f≡0)? Then return unity. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
genPrimes: arg x; @.=; @.1=2; @.2=3; #.=.; #=2; s.#=@.#**2
do j=@.# +2 by 2 to x /*only find odd primes from here on. */
do p=2 while s.p<=j /*divide by some known low odd primes. */
if j//@.p==0 then iterate j /*Is J divisible by X? Then ¬ prime.*/
end /*p*/ /* [↓] a prime (J) has been found. */
#=#+1; @.#=j; #.j=1; s.#=j*j /*bump prime count, and also assign ···*/
end /*j*/ /* ··· the # of factors, prime, prime².*/
return /* [↑] not an optimal prime generator.*/
- output when using the input of: 20 16
The highest prime computed: 655357 (under the limit of 655360). 1─almost (20) primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 2─almost (20) primes: 4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 3─almost (20) primes: 8 12 18 20 27 28 30 42 44 45 50 52 63 66 68 70 75 76 78 92 4─almost (20) primes: 16 24 36 40 54 56 60 81 84 88 90 100 104 126 132 135 136 140 150 152 5─almost (20) primes: 32 48 72 80 108 112 120 162 168 176 180 200 208 243 252 264 270 272 280 300 6─almost (20) primes: 64 96 144 160 216 224 240 324 336 352 360 400 416 486 504 528 540 544 560 600 7─almost (20) primes: 128 192 288 320 432 448 480 648 672 704 720 800 832 972 1008 1056 1080 1088 1120 1200 8─almost (20) primes: 256 384 576 640 864 896 960 1296 1344 1408 1440 1600 1664 1944 2016 2112 2160 2176 2240 2400 9─almost (20) primes: 512 768 1152 1280 1728 1792 1920 2592 2688 2816 2880 3200 3328 3888 4032 4224 4320 4352 4480 4800 10─almost (20) primes: 1024 1536 2304 2560 3456 3584 3840 5184 5376 5632 5760 6400 6656 7776 8064 8448 8640 8704 8960 9600 11─almost (20) primes: 2048 3072 4608 5120 6912 7168 7680 10368 10752 11264 11520 12800 13312 15552 16128 16896 17280 17408 17920 19200 12─almost (20) primes: 4096 6144 9216 10240 13824 14336 15360 20736 21504 22528 23040 25600 26624 31104 32256 33792 34560 34816 35840 38400 13─almost (20) primes: 8192 12288 18432 20480 27648 28672 30720 41472 43008 45056 46080 51200 53248 62208 64512 67584 69120 69632 71680 76800 14─almost (20) primes: 16384 24576 36864 40960 55296 57344 61440 82944 86016 90112 92160 102400 106496 124416 129024 135168 138240 139264 143360 153600 15─almost (20) primes: 32768 49152 73728 81920 110592 114688 122880 165888 172032 180224 184320 204800 212992 248832 258048 270336 276480 278528 286720 307200 16─almost (20) primes: 65536 98304 147456 163840 221184 229376 245760 331776 344064 360448 368640 409600 425984 497664 516096 540672 552960 557056 573440 614400 0.088 seconds (Regina)
Yes, this is very fast. But both Version 1 and Version 2 are highly optimized for this specific task. They take advantage from the fact that we calculate for k = 1,2,3 and so on, thus use patterns in the generated primes and can pre generate and use a list of primes. In Version 2 you see also several several hardcoded numbers, specific for this task. And by the way, both programs are rather complicated.
A more standard approach, using a procedure Factors (see Prime decomposition), follows.
Version 4 Standard procedures
Libraries: How to use
Library: Numbers
Library: Functions
Library: Settings
Library: Abend
include Settings
say version; say 'k-Almost primes'; say
arg n k m
say 'Direct approach using Factors'
numeric digits 16
if n = '' then
n = 10
if k = '' then
k = 5
/* Maximum number to examine */
if m = '' then
m = 180
call Time('r')
/* Collect almost primes */
ap. = 0
do i = 2 to m
f = Factors(i); ap.f.0 = ap.f.0+1
ap = ap.f.0; ap.f.ap = i
end
/* Show results */
do i = 1 to k
call Charout ,'k='i': '
do j = 1 to n
if ap.i.j > 0 then do
call Charout ,ap.i.j' '
end
end
say
end
say Format(Time('e'),,3) 'seconds'
exit
include Numbers
include Functions
include Abend
The maximum number is parameter here, but may be estimated from n and k, as in Versions 1 and 2.
- Output default parameters:
REXX-Regina_3.9.6(MT) 5.00 29 Apr 2024 k-Almost primes Direct approach using Factors k=1: 2 3 5 7 11 13 17 19 23 29 k=2: 4 6 9 10 14 15 21 22 25 26 k=3: 8 12 18 20 27 28 30 42 44 45 k=4: 16 24 36 40 54 56 60 81 84 88 k=5: 32 48 72 80 108 112 120 162 168 176 0.003 seconds
- Output parameters 20 12 39000:
REXX-Regina_3.9.6(MT) 5.00 29 Apr 2024 16 digits k-Almost primes Direct approach using Factors k=1: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 k=2: 4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 k=3: 8 12 18 20 27 28 30 42 44 45 50 52 63 66 68 70 75 76 78 92 k=4: 16 24 36 40 54 56 60 81 84 88 90 100 104 126 132 135 136 140 150 152 k=5: 32 48 72 80 108 112 120 162 168 176 180 200 208 243 252 264 270 272 280 300 k=6: 64 96 144 160 216 224 240 324 336 352 360 400 416 486 504 528 540 544 560 600 k=7: 128 192 288 320 432 448 480 648 672 704 720 800 832 972 1008 1056 1080 1088 1120 1200 k=8: 256 384 576 640 864 896 960 1296 1344 1408 1440 1600 1664 1944 2016 2112 2160 2176 2240 2400 k=9: 512 768 1152 1280 1728 1792 1920 2592 2688 2816 2880 3200 3328 3888 4032 4224 4320 4352 4480 4800 k=10: 1024 1536 2304 2560 3456 3584 3840 5184 5376 5632 5760 6400 6656 7776 8064 8448 8640 8704 8960 9600 k=11: 2048 3072 4608 5120 6912 7168 7680 10368 10752 11264 11520 12800 13312 15552 16128 16896 17280 17408 17920 19200 k=12: 4096 6144 9216 10240 13824 14336 15360 20736 21504 22528 23040 25600 26624 31104 32256 33792 34560 34816 35840 38400 0.885 seconds
- Output parameters 20 16 615000:
REXX-Regina_3.9.6(MT) 5.00 29 Apr 2024 16 digits k-Almost primes Direct approach using Factors k=1: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 k=2: 4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 k=3: 8 12 18 20 27 28 30 42 44 45 50 52 63 66 68 70 75 76 78 92 k=4: 16 24 36 40 54 56 60 81 84 88 90 100 104 126 132 135 136 140 150 152 k=5: 32 48 72 80 108 112 120 162 168 176 180 200 208 243 252 264 270 272 280 300 k=6: 64 96 144 160 216 224 240 324 336 352 360 400 416 486 504 528 540 544 560 600 k=7: 128 192 288 320 432 448 480 648 672 704 720 800 832 972 1008 1056 1080 1088 1120 1200 k=8: 256 384 576 640 864 896 960 1296 1344 1408 1440 1600 1664 1944 2016 2112 2160 2176 2240 2400 k=9: 512 768 1152 1280 1728 1792 1920 2592 2688 2816 2880 3200 3328 3888 4032 4224 4320 4352 4480 4800 k=10: 1024 1536 2304 2560 3456 3584 3840 5184 5376 5632 5760 6400 6656 7776 8064 8448 8640 8704 8960 9600 k=11: 2048 3072 4608 5120 6912 7168 7680 10368 10752 11264 11520 12800 13312 15552 16128 16896 17280 17408 17920 19200 k=12: 4096 6144 9216 10240 13824 14336 15360 20736 21504 22528 23040 25600 26624 31104 32256 33792 34560 34816 35840 38400 k=13: 8192 12288 18432 20480 27648 28672 30720 41472 43008 45056 46080 51200 53248 62208 64512 67584 69120 69632 71680 76800 k=14: 16384 24576 36864 40960 55296 57344 61440 82944 86016 90112 92160 102400 106496 124416 129024 135168 138240 139264 143360 153600 k=15: 32768 49152 73728 81920 110592 114688 122880 165888 172032 180224 184320 204800 212992 248832 258048 270336 276480 278528 286720 307200 k=16: 65536 98304 147456 163840 221184 229376 245760 331776 344064 360448 368640 409600 425984 497664 516096 540672 552960 557056 573440 614400 25.129 seconds
Not too bad! By the way, Version 3 can also generate lists of almost prime over other number ranges. Say you change the first do in 'do 1000000 to m' and run as follows, you get
- Output parameters 10 10 1002000:
REXX-Regina_3.9.6(MT) 5.00 29 Apr 2024 16 digits k-Almost primes Direct approach using Factors k=1: 1000003 1000033 1000037 1000039 1000081 1000099 1000117 1000121 1000133 1000151 k=2: 1000001 1000007 1000009 1000011 1000015 1000018 1000019 1000021 1000023 1000031 k=3: 1000002 1000006 1000013 1000014 1000022 1000028 1000029 1000030 1000043 1000046 k=4: 1000005 1000010 1000012 1000017 1000024 1000027 1000034 1000038 1000041 1000042 k=5: 1000004 1000016 1000025 1000035 1000036 1000044 1000056 1000060 1000062 1000072 k=6: 1000020 1000026 1000040 1000048 1000050 1000065 1000076 1000090 1000096 1000100 k=7: 1000125 1000152 1000176 1000200 1000256 1000352 1000368 1000404 1000428 1000431 k=8: 1000008 1000032 1000128 1000272 1000296 1000400 1000416 1000440 1000500 1000560 k=9: 1000064 1000080 1000160 1000192 1000224 1000350 1000480 1000640 1000832 1000896 k=10: 1000320 1000384 1000704 1000800 1001160 1001280 1001376 1001600 1001664 1001952 0.116 seconds
Ring
for ap = 1 to 5
see "k = " + ap + ":"
aList = []
for n = 1 to 200
num = 0
for nr = 1 to n
if n%nr=0 and isPrime(nr)=1
num = num + 1
pr = nr
while true
pr = pr * nr
if n%pr = 0
num = num + 1
else exit ok
end ok
next
if (ap = 1 and isPrime(n) = 1) or (ap > 1 and num = ap)
add(aList, n)
if len(aList)=10 exit ok ok
next
for m = 1 to len(aList)
see " " + aList[m]
next
see nl
next
func isPrime num
if (num <= 1) return 0 ok
if (num % 2 = 0 and num != 2) return 0 ok
for i = 3 to floor(num / 2) -1 step 2
if (num % i = 0) return 0 ok
next
return 1Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
RPL
| RPL code | Comment |
|---|---|
≪ → k
≪ 0 1 SF
2 3 PICK FOR j
WHILE OVER j MOD NOT REPEAT
IF DUP k == THEN 1 CF OVER 'j' STO END
1 +
SWAP j / SWAP
END
NEXT
k == 1 FS? AND SWAP DROP
≫ ≫ 'KPRIM' STO
≪ 5 1 FOR k
{ } 2
WHILE OVER SIZE 10 < REPEAT
IF DUP k KPRIM THEN SWAP OVER + SWAP END
1 + END DROP
-1 STEP
≫ 'TASK' STO
|
KPRIM ( n k → boolean )
Dim f As Integer = 0
For i As Integer = 2 To n
While n Mod i = 0
If f = k Then Return false
f += 1
n \= i
Wend
Next
Return f = k
End Function
|
- Output:
5 : { 32 48 72 80 108 112 120 162 168 176 }
4 : { 16 24 36 40 54 56 60 81 84 88 }
3 : { 8 12 18 20 27 28 30 42 44 45 }
2 : { 4 6 9 10 14 15 21 22 25 26 }
1 : { 2 3 5 7 9 11 13 17 19 23 29 }
Ruby
require 'prime'
def almost_primes(k=2)
return to_enum(:almost_primes, k) unless block_given?
1.step {|n| yield n if n.prime_division.sum( &:last ) == k }
end
(1..5).each{|k| puts almost_primes(k).take(10).join(", ")}
- Output:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 4, 6, 9, 10, 14, 15, 21, 22, 25, 26 8, 12, 18, 20, 27, 28, 30, 42, 44, 45 16, 24, 36, 40, 54, 56, 60, 81, 84, 88 32, 48, 72, 80, 108, 112, 120, 162, 168, 176
require 'prime'
p ar = pr = Prime.take(10)
4.times{p ar = ar.product(pr).map{|(a,b)| a*b}.uniq.sort.take(10)}
- Output:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29] [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Rust
fn is_kprime(n: u32, k: u32) -> bool {
let mut primes = 0;
let mut f = 2;
let mut rem = n;
while primes < k && rem > 1{
while (rem % f) == 0 && rem > 1{
rem /= f;
primes += 1;
}
f += 1;
}
rem == 1 && primes == k
}
struct KPrimeGen {
k: u32,
n: u32,
}
impl Iterator for KPrimeGen {
type Item = u32;
fn next(&mut self) -> Option<u32> {
self.n += 1;
while !is_kprime(self.n, self.k) {
self.n += 1;
}
Some(self.n)
}
}
fn kprime_generator(k: u32) -> KPrimeGen {
KPrimeGen {k: k, n: 1}
}
fn main() {
for k in 1..6 {
println!("{}: {:?}", k, kprime_generator(k).take(10).collect::<Vec<_>>());
}
}
- Output:
1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] 2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] 3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] 4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] 5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Scala
def isKPrime(n: Int, k: Int, d: Int = 2): Boolean = (n, k, d) match {
case (n, k, _) if n == 1 => k == 0
case (n, _, d) if n % d == 0 => isKPrime(n / d, k - 1, d)
case (_, _, _) => isKPrime(n, k, d + 1)
}
def kPrimeStream(k: Int): Stream[Int] = {
def loop(n: Int): Stream[Int] =
if (isKPrime(n, k)) n #:: loop(n+ 1)
else loop(n + 1)
loop(2)
}
for (k <- 1 to 5) {
println( s"$k: [${ kPrimeStream(k).take(10) mkString " " }]" )
}
- Output:
1: [2 3 5 7 11 13 17 19 23 29] 2: [4 6 9 10 14 15 21 22 25 26] 3: [8 12 18 20 27 28 30 42 44 45] 4: [16 24 36 40 54 56 60 81 84 88] 5: [32 48 72 80 108 112 120 162 168 176]
Seed7
$ include "seed7_05.s7i";
const func boolean: kprime (in var integer: number, in integer: k) is func
result
var boolean: kprime is FALSE;
local
var integer: p is 2;
var integer: f is 0;
begin
while f < k and p * p <= number do
while number rem p = 0 do
number := number div p;
incr(f);
end while;
incr(p);
end while;
kprime := f + ord(number > 1) = k;
end func;
const proc: main is func
local
var integer: k is 0;
var integer: number is 0;
var integer: count is 0;
begin
for k range 1 to 5 do
write("k = " <& k <& ":");
count := 0;
for number range 2 to integer.last until count >= 10 do
if kprime(number, k) then
write(" " <& number);
incr(count);
end if;
end for;
writeln;
end for;
end func;- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
SequenceL
import <Utilities/Conversion.sl>;
import <Utilities/Sequence.sl>;
main(args(2)) :=
let
result := firstNKPrimes(1 ... 5, 10);
output[i] := "k = " ++ intToString(i) ++ ": " ++ delimit(intToString(result[i]), ' ');
in
delimit(output, '\n');
firstNKPrimes(k, N) := firstNKPrimesHelper(k, N, 2, []);
firstNKPrimesHelper(k, N, current, result(1)) :=
let
newResult := result when not isKPrime(k, current) else result ++ [current];
in
result when size(result) = N
else
firstNKPrimesHelper(k, N, current + 1, newResult);
isKPrime(k, n) := size(primeFactorization(n)) = k;Using Prime Decomposition Solution [1]
- Output:
main.exe "k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176"
Sidef
Efficient algorithm for generating all the k-almost prime numbers in a given range [a,b]:
func almost_primes(a, b, k) {
a = max(2**k, a)
var arr = []
func (m, lo, k) {
var hi = idiv(b,m).iroot(k)
if (k == 1) {
lo = max(lo, idiv_ceil(a, m))
each_prime(lo, hi, {|p|
arr << m*p
})
return nil
}
each_prime(lo, hi, {|p|
var t = m*p
var u = idiv_ceil(a, t)
var v = idiv(b, t)
next if (u > v)
__FUNC__(t, p, k-1)
})
}(1, 2, k)
return arr.sort
}
for k in (1..5) {
var (x=10, lo=1, hi=2)
var arr = []
loop {
arr += almost_primes(lo, hi, k)
break if (arr.len >= x)
lo = hi+1
hi = 2*lo
}
say arr.first(x)
}
- Output:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29] [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Also built-in:
for k in (1..5) {
var x = 10
say k.almost_primes(x.nth_almost_prime(k))
}
(same output as above)
Swift
struct KPrimeGen: Sequence, IteratorProtocol {
let k: Int
private(set) var n: Int
private func isKPrime() -> Bool {
var primes = 0
var f = 2
var rem = n
while primes < k && rem > 1 {
while rem % f == 0 && rem > 1 {
rem /= f
primes += 1
}
f += 1
}
return rem == 1 && primes == k
}
mutating func next() -> Int? {
n += 1
while !isKPrime() {
n += 1
}
return n
}
}
for k in 1..<6 {
print("\(k): \(Array(KPrimeGen(k: k, n: 1).lazy.prefix(10)))")
}
- Output:
1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] 2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] 3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] 4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] 5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
Tcl
package require Tcl 8.6
package require math::numtheory
proc firstNprimes n {
for {set result {};set i 2} {[llength $result] < $n} {incr i} {
if {[::math::numtheory::isprime $i]} {
lappend result $i
}
}
return $result
}
proc firstN_KalmostPrimes {n k} {
set p [firstNprimes $n]
set i [lrepeat $k 0]
set c {}
while true {
dict set c [::tcl::mathop::* {*}[lmap j $i {lindex $p $j}]] ""
for {set x 0} {$x < $k} {incr x} {
lset i $x [set xx [expr {([lindex $i $x] + 1) % $n}]]
if {$xx} break
}
if {$x == $k} break
}
return [lrange [lsort -integer [dict keys $c]] 0 [expr {$n - 1}]]
}
for {set K 1} {$K <= 5} {incr K} {
puts "$K => [firstN_KalmostPrimes 10 $K]"
}
- Output:
1 => 2 3 5 7 11 13 17 19 23 29 2 => 4 6 9 10 14 15 21 22 25 26 3 => 8 12 18 20 27 28 30 42 44 45 4 => 16 24 36 40 54 56 60 81 84 88 5 => 32 48 72 80 108 112 120 162 168 176
TypeScript
// Almost prime
function isKPrime(n: number, k: number): bool {
var f = 0;
for (var i = 2; i <= n; i++)
while (n % i == 0) {
if (f == k)
return false;
++f;
n = Math.floor(n / i);
}
return f == k;
}
for (var k = 1; k <= 5; k++) {
process.stdout.write(`k = ${k}:`);
var i = 2, c = 0;
while (c < 10) {
if (isKPrime(i, k)) {
process.stdout.write(" " + i.toString().padStart(3, ' '));
++c;
}
++i;
}
console.log();
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
VBA
Private Function kprime(ByVal n As Integer, k As Integer) As Boolean
Dim p As Integer, factors As Integer
p = 2
factors = 0
Do While factors < k And p * p <= n
Do While n Mod p = 0
n = n / p
factors = factors + 1
Loop
p = p + 1
Loop
factors = factors - (n > 1) 'true=-1
kprime = factors = k
End Function
Private Sub almost_primeC()
Dim nextkprime As Integer, count As Integer
Dim k As Integer
For k = 1 To 5
Debug.Print "k ="; k; ":";
nextkprime = 2
count = 0
Do While count < 10
If kprime(nextkprime, k) Then
Debug.Print " "; Format(CStr(nextkprime), "@@@@@");
count = count + 1
End If
nextkprime = nextkprime + 1
Loop
Debug.Print
Next k
End Sub
- Output:
k = 1 : 2 3 5 7 11 13 17 19 23 29 k = 2 : 4 6 9 10 14 15 21 22 25 26 k = 3 : 8 12 18 20 27 28 30 42 44 45 k = 4 : 16 24 36 40 54 56 60 81 84 88 k = 5 : 32 48 72 80 108 112 120 162 168 176
VBScript
Repurposed the VBScript code for the Prime Decomposition task.
For k = 1 To 5
count = 0
increment = 1
WScript.StdOut.Write "K" & k & ": "
Do Until count = 10
If PrimeFactors(increment) = k Then
WScript.StdOut.Write increment & " "
count = count + 1
End If
increment = increment + 1
Loop
WScript.StdOut.WriteLine
Next
Function PrimeFactors(n)
PrimeFactors = 0
arrP = Split(ListPrimes(n)," ")
divnum = n
Do Until divnum = 1
For i = 0 To UBound(arrP)-1
If divnum = 1 Then
Exit For
ElseIf divnum Mod arrP(i) = 0 Then
divnum = divnum/arrP(i)
PrimeFactors = PrimeFactors + 1
End If
Next
Loop
End Function
Function IsPrime(n)
If n = 2 Then
IsPrime = True
ElseIf n <= 1 Or n Mod 2 = 0 Then
IsPrime = False
Else
IsPrime = True
For i = 3 To Int(Sqr(n)) Step 2
If n Mod i = 0 Then
IsPrime = False
Exit For
End If
Next
End If
End Function
Function ListPrimes(n)
ListPrimes = ""
For i = 1 To n
If IsPrime(i) Then
ListPrimes = ListPrimes & i & " "
End If
Next
End Function
- Output:
K1: 2 3 5 7 11 13 17 19 23 29 K2: 4 6 9 10 14 15 21 22 25 26 K3: 8 12 18 20 27 28 30 42 44 45 K4: 16 24 36 40 54 56 60 81 84 88 K5: 32 48 72 80 108 112 120 162 168 176
V (Vlang)
fn k_prime(n int, k int) bool {
mut nf := 0
mut nn := n
for i in 2..nn + 1 {
for nn % i == 0 {
if nf == k {return false}
nf++
nn /= i
}
}
return nf == k
}
fn gen(k int, n int) []int {
mut r := []int{len:n}
mut nx := 2
for i in 0..n {
for !k_prime(nx, k) {nx++}
r[i] = nx
nx++
}
return r
}
fn main(){
for k in 1..6 {println('$k ${gen(k,10)}')}
}- Output:
1 [2 3 5 7 11 13 17 19 23 29] 2 [4 6 9 10 14 15 21 22 25 26] 3 [8 12 18 20 27 28 30 42 44 45] 4 [16 24 36 40 54 56 60 81 84 88] 5 [32 48 72 80 108 112 120 162 168 176]
Wren
var kPrime = Fn.new { |n, k|
var nf = 0
var i = 2
while (i <= n) {
while (n%i == 0) {
if (nf == k) return false
nf = nf + 1
n = (n/i).floor
}
i = i + 1
}
return nf == k
}
var gen = Fn.new { |k, n|
var r = List.filled(n, 0)
n = 2
for (i in 0...r.count) {
while (!kPrime.call(n, k)) n = n + 1
r[i] = n
n = n + 1
}
return r
}
for (k in 1..5) System.print("%(k) %(gen.call(k, 10))")
- Output:
1 [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] 2 [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] 3 [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] 4 [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] 5 [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
XPL0
func Factors(N); \Return number of (prime) factors in N
int N, F, C;
[C:= 0; F:= 2;
repeat if rem(N/F) = 0 then
[C:= C+1;
N:= N/F;
]
else F:= F+1;
until F > N;
return C;
];
int K, C, N;
[for K:= 1 to 5 do
[C:= 0;
N:= 2;
IntOut(0, K); Text(0, ": ");
loop [if Factors(N) = K then
[IntOut(0, N); ChOut(0, ^ );
C:= C+1;
if C >= 10 then quit;
];
N:= N+1;
];
CrLf(0);
];
]- Output:
1: 2 3 5 7 11 13 17 19 23 29 2: 4 6 9 10 14 15 21 22 25 26 3: 8 12 18 20 27 28 30 42 44 45 4: 16 24 36 40 54 56 60 81 84 88 5: 32 48 72 80 108 112 120 162 168 176
zkl
Using the prime generator from task Extensible prime generator#zkl.
Can't say I entirely understand this algorithm. Uses list comprehension to calculate the outer/tensor product (p10 ⊗ ar).
primes:=Utils.Generator(Import("sieve").postponed_sieve);
(p10:=ar:=primes.walk(10)).println();
do(4){
(ar=([[(x,y);ar;p10;'*]] : Utils.Helpers.listUnique(_).sort()[0,10])).println();
}- Output:
L(2,3,5,7,11,13,17,19,23,29) L(4,6,9,10,14,15,21,22,25,26) L(8,12,18,20,27,28,30,42,44,45) L(16,24,36,40,54,56,60,81,84,88) L(32,48,72,80,108,112,120,162,168,176)
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