# Extensible prime generator

**Extensible prime generator**

You are encouraged to solve this task according to the task description, using any language you may know.

- Task

Write a generator of prime numbers, in order, that will automatically adjust to accommodate the generation of any reasonably high prime.

The routine should demonstrably rely on either:

- Being based on an open-ended counter set to count without upper limit other than system or programming language limits. In this case, explain where this counter is in the code.
- Being based on a limit that is extended automatically. In this case, choose a small limit that ensures the limit will be passed when generating some of the values to be asked for below.
- If other methods of creating an extensible prime generator are used, the algorithm's means of extensibility/lack of limits should be stated.

The routine should be used to:

- Show the first twenty primes.
- Show the primes between 100 and 150.
- Show the
*number*of primes between 7,700 and 8,000. - Show the 10,000th prime.

Show output on this page.

**Note:** You may reference code already on this site if it is written to be imported/included, then only the code necessary for import and the performance of this task need be shown. (It is also important to leave a forward link on the referenced tasks entry so that later editors know that the code is used for multiple tasks).

**Note 2:** If a languages in-built prime generator is extensible or is guaranteed to generate primes up to a system limit, (2^{31} or memory overflow for example), then this may be used as long as an explanation of the limits of the prime generator is also given. (Which may include a link to/excerpt from, language documentation).

**Note 3:**The task is written so it may be useful in solving the task Emirp primes as well as others (depending on its efficiency).

- Reference

- Prime Numbers. Website with large count of primes.

## Ada

The solution is based on an open-ended counter, named "Current" counting up to the limit from the Compiler, namely 2**63-1.

The solution uses the package Miller_Rabin from the Miller-Rabin primality test. When using the gnat Ada compiler, the largest integer we can deal with is 2**63-1. For anything larger, we could use a big-num package.

```
with Ada.Text_IO, Miller_Rabin;
procedure Prime_Gen is
type Num is range 0 .. 2**63-1; -- maximum for the gnat Ada compiler
MR_Iterations: constant Positive := 25;
-- the probability Pr[Is_Prime(N, MR_Iterations) = Probably_Prime]
-- is 1 for prime N and < 4**(-MR_Iterations) for composed N
function Next(P: Num) return Num is
N: Num := P+1;
package MR is new Miller_Rabin(Num); use MR;
begin
while not (Is_Prime(N, MR_Iterations) = Probably_Prime) loop
N := N + 1;
end loop;
return N;
end Next;
Current: Num;
Count: Num := 0;
begin
-- show the first twenty primes
Ada.Text_IO.Put("First 20 primes:");
Current := 1;
for I in 1 .. 20 loop
Current := Next(Current);
Ada.Text_IO.Put(Num'Image(Current));
end loop;
Ada.Text_IO.New_Line;
-- show the primes between 100 and 150
Ada.Text_IO.Put("Primes between 100 and 150:");
Current := 99;
loop
Current := Next(Current);
exit when Current > 150;
Ada.Text_IO.Put(Num'Image(Current));
end loop;
Ada.Text_IO.New_Line;
-- count primes between 7700 and 8000
Ada.Text_IO.Put("Number of primes between 7700 and 8000:");
Current := 7699;
loop
Current := Next(Current);
exit when Current > 8000;
Count := Count + 1;
end loop;
Ada.Text_IO.Put_Line(Num'Image(Count));
Count := 10;
Ada.Text_IO.Put_Line("Print the K_i'th prime, for $K=10**i:");
begin
loop
Current := 1;
for I in 1 .. Count loop
Current := Next(Current);
end loop;
Ada.Text_IO.Put(Num'Image(Count) & "th prime:" &
Num'Image(Current));
Count := Count * 10;
end loop;
exception
when Constraint_Error =>
Ada.Text_IO.Put_Line(" can't compute the" & Num'Image(Count) &
"th prime:");
end;
end;
```

- Output:

First 20 primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 Primes between 100 and 150: 101 103 107 109 113 127 131 137 139 149 Number of primes between 7700 and 8000: 30 Print the K_i'th prime, for $K=10**i: 10th prime: 29 100th prime: 541 1000th prime: 7919 10000th prime: 104729 100000th prime: 1299709 1000000th prime: 15485863

(The program has been stopped after running several days.)

## AutoHotkey

```
SetBatchLines, -1
p := 1 ;p functions as the counter
Loop, 10000 {
p := NextPrime(p)
if (A_Index < 21)
a .= p ", "
if (p < 151 && p > 99)
b .= p ", "
if (p < 8001 && p > 7699)
c++
}
MsgBox, % "First twenty primes: " RTrim(a, ", ")
. "`nPrimes between 100 and 150: " RTrim(b, ", ")
. "`nNumber of primes between 7,700 and 8,000: " RTrim(c, ", ")
. "`nThe 10,000th prime: " p
NextPrime(n) {
Loop
if (IsPrime(++n))
return n
}
IsPrime(n) {
if (n < 2)
return, 0
else if (n < 4)
return, 1
else if (!Mod(n, 2))
return, 0
else if (n < 9)
return 1
else if (!Mod(n, 3))
return, 0
else {
r := Floor(Sqrt(n))
f := 5
while (f <= r) {
if (!Mod(n, f))
return, 0
if (!Mod(n, (f + 2)))
return, 0
f += 6
}
return, 1
}
}
```

- Output:

First twenty primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 Primes between 100 and 150: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149 Number of primes between 7,700 and 8,000: 30 The 10,000th prime: 104729

## BASIC

### FreeBASIC

This program uses the Sieve Of Eratosthenes which is not very efficient for large primes but is quick enough for present purposes.

The size of the sieve array (of type Boolean) is calculated as 20 times the number of primes required which is big enough to compute up to 50 million primes (a sieve size of 1 billion bytes) which takes under 50 seconds on my i3 @ 2.13 GHz. I've limited the procedure to this but it should certainly be possible to use a much higher figure without running out of memory.

It would also be possible to use a more efficient algorithm to compute the optimal sieve size for smaller numbers of primes but this will suffice for now.

```
' FB 1.05.0
Enum SieveLimitType
number
between
countBetween
End Enum
Sub printPrimes(low As Integer, high As Integer, slt As SieveLimitType)
If high < low OrElse low < 1 Then Return ' too small
If slt <> number AndAlso slt <> between AndAlso slt <> countBetween Then Return
If slt <> number AndAlso (low < 2 OrElse high < 2) Then Return
If slt <> number AndAlso high > 1000000000 Then Return ' too big
If slt = number AndAlso high > 50000000 Then Return ' too big
Dim As Integer n
If slt = number Then
n = 20 * high '' big enough to accomodate 50 million primes to which this procedure is limited
Else
n = high
End If
Dim a(2 To n) As Boolean '' only uses 1 byte per element
For i As Integer = 2 To n : a(i) = True : Next '' set all elements to True to start with
Dim As Integer p = 2, q
' mark non-prime numbers by setting the corresponding array element to False
Do
For j As Integer = p * p To n Step p
a(j) = False
Next j
' look for next True element in array after 'p'
q = 0
For j As Integer = p + 1 To Sqr(n)
If a(j) Then
q = j
Exit For
End If
Next j
If q = 0 Then Exit Do
p = q
Loop
Select Case As Const slt
Case number
Dim count As Integer = 0
For i As Integer = 2 To n
If a(i) Then
count += 1
If count >= low AndAlso count <= high Then
Print i; " ";
End If
If count = high Then Exit Select
End If
Next
Case between
For i As Integer = low To high
If a(i) Then
Print i; " ";
End if
Next
Case countBetween
Dim count As Integer = 0
For i As Integer = low To high
If a(i) Then count += 1
Next
Print count;
End Select
Print
End Sub
Print "The first 20 primes are :"
Print
printPrimes(1, 20, number)
Print
Print "The primes between 100 and 150 are :"
Print
printPrimes(100, 150, between)
Print
Print "The number of primes between 7700 and 8000 is :";
printPrimes(7700, 8000, countBetween)
Print
Print "The 10000th prime is :";
Dim t As Double = timer
printPrimes(10000, 10000, number)
Print "Computed in "; CInt((timer - t) * 1000 + 0.5); " ms"
Print
Print "The 1000000th prime is :";
t = timer
printPrimes(1000000, 1000000, number)
Print "Computed in ";CInt((timer - t) * 1000 + 0.5); " ms"
Print
Print "The 50000000th prime is :";
t = timer
printPrimes(50000000, 50000000, number)
Print "Computed in ";CInt((timer - t) * 1000 + 0.5); " ms"
Print
Print "Press any key to quit"
Sleep
```

- Output:

The first 20 primes are : 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 The primes between 100 and 150 are : 101 103 107 109 113 127 131 137 139 149 The number of primes between 7700 and 8000 is : 30 The 10000th prime is : 104729 Computed in 8 ms The 1000000th prime is : 15485863 Computed in 775 ms The 50000000th prime is : 982451653 Computed in 46703 ms

## BQN

This implementation uses a simple segmented sieve similar to the one in Sieve of Eratosthenes. In order to match the task most closely, it returns a generator function (implemented as a closure) that outputs another prime on each call, using an underlying `primes`

array that's extended as necessary. Working with one prime at a time is inefficient in an array language, and the given tasks can be solved more quickly using functions from bqn-libs primes.bqn, which also uses a more complicated and faster underlying sieve.

```
# Function that returns a new prime generator
PrimeGen ← {𝕤
i ← 0 # Counter: index of next prime to be output
primes ← ↕0
next ← 2
Sieve ← { p 𝕊 i‿n:
E ← {↕∘⌈⌾(((𝕩|-i)+𝕩×⊢)⁼)n-i} # Indices of multiples of 𝕩
i + / (1⥊˜n-i) E⊸{0¨⌾(𝕨⊸⊏)𝕩}´ p # Primes in segment [i,n)
}
{𝕤
{ i=≠primes ? # Extend if required
next ↩ ((2⋆24)⊸+ ⌊ ×˜) old←next # Sieve at most 16M new entries
primes ∾↩ (primes(⍋↑⊣)√next) Sieve old‿next
;@}
(i+↩1) ⊢ i⊑primes
}
}
_w_←{𝔽⍟𝔾∘𝔽_𝕣_𝔾∘𝔽⍟𝔾𝕩} # Looping utility for the session below
```

- Output:

```
pg ← PrimeGen@
(function block)
PG¨ ↕20
⟨ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ⟩
{p←↕0 ⋄ PG∘{ p∾↩𝕩}_w_(<⟜ 150) PG _w_(<⟜ 100)0 ⋄ p}
⟨ 101 103 107 109 113 127 131 137 139 149 ⟩
{p←0 ⋄ PG∘{𝕤⋄p+↩1}_w_(<⟜8000) PG _w_(<⟜7700)0 ⋄ p}
30
(PrimeGen@)⍟1e4 @ # Reset the count with a new generator
104729
```

## C

Extends the list of primes by sieving more chunks of integers. There's no serious optimizations. The code can calculate all 32-bit primes in some seconds, and will overflow beyond that.

```
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#define CHUNK_BYTES (32 << 8)
#define CHUNK_SIZE (CHUNK_BYTES << 6)
int field[CHUNK_BYTES];
#define GET(x) (field[(x)>>6] & 1<<((x)>>1&31))
#define SET(x) (field[(x)>>6] |= 1<<((x)>>1&31))
typedef unsigned uint;
typedef struct {
uint *e;
uint cap, len;
} uarray;
uarray primes, offset;
void push(uarray *a, uint n)
{
if (a->len >= a->cap) {
if (!(a->cap *= 2)) a->cap = 16;
a->e = realloc(a->e, sizeof(uint) * a->cap);
}
a->e[a->len++] = n;
}
uint low;
void init(void)
{
uint p, q;
unsigned char f[1<<16];
memset(f, 0, sizeof(f));
push(&primes, 2);
push(&offset, 0);
for (p = 3; p < 1<<16; p += 2) {
if (f[p]) continue;
for (q = p*p; q < 1<<16; q += 2*p) f[q] = 1;
push(&primes, p);
push(&offset, q);
}
low = 1<<16;
}
void sieve(void)
{
uint i, p, q, hi, ptop;
if (!low) init();
memset(field, 0, sizeof(field));
hi = low + CHUNK_SIZE;
ptop = sqrt(hi) * 2 + 1;
for (i = 1; (p = primes.e[i]*2) < ptop; i++) {
for (q = offset.e[i] - low; q < CHUNK_SIZE; q += p)
SET(q);
offset.e[i] = q + low;
}
for (p = 1; p < CHUNK_SIZE; p += 2)
if (!GET(p)) push(&primes, low + p);
low = hi;
}
int main(void)
{
uint i, p, c;
while (primes.len < 20) sieve();
printf("First 20:");
for (i = 0; i < 20; i++)
printf(" %u", primes.e[i]);
putchar('\n');
while (primes.e[primes.len-1] < 150) sieve();
printf("Between 100 and 150:");
for (i = 0; i < primes.len; i++) {
if ((p = primes.e[i]) >= 100 && p < 150)
printf(" %u", primes.e[i]);
}
putchar('\n');
while (primes.e[primes.len-1] < 8000) sieve();
for (i = c = 0; i < primes.len; i++)
if ((p = primes.e[i]) >= 7700 && p < 8000) c++;
printf("%u primes between 7700 and 8000\n", c);
for (c = 10; c <= 100000000; c *= 10) {
while (primes.len < c) sieve();
printf("%uth prime: %u\n", c, primes.e[c-1]);
}
return 0;
}
```

- Output:

First 20: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 Between 100 and 150: 101 103 107 109 113 127 131 137 139 149 30 primes between 7700 and 8000 10th prime: 29 100th prime: 541 1000th prime: 7919 10000th prime: 104729 100000th prime: 1299709 1000000th prime: 15485863 10000000th prime: 179424673 100000000th prime: 2038074743

### Alternative version based on The Genuine Sieve of Eratosthenes by Melissa O'Neil

It uses the Pairing Heap w/ generic data types from task Priority queue#C. Some notes:

- It uses a wheel to skip over multiples of 2, 3, 5, and 7.
- since we store the square of every prime found, the use of long int (64 bit in GCC) is required for a sieve of any non-trivial size.
- ~100,000,000 primes is about the limits of this algorithm. This version is not as idiomatic to C as a page-segmented sieve would be.

```
#include <stdio.h>
#include <stdlib.h>
#include "pairheap.h"
int wheel2357[48] = {
10, 2, 4, 2, 4, 6, 2, 6,
4, 2, 4, 6, 6, 2, 6, 4,
2, 6, 4, 6, 8, 4, 2, 4,
2, 4, 8, 6, 4, 6, 2, 4,
6, 2, 6, 6, 4, 2, 4, 6,
2, 6, 4, 2, 4, 2, 10, 2,
};
typedef struct { // elements in the priority queue
pq_node_t hd;
int offset; // index to skip value in 2,3,5,7 wheel
long int base_prime;
} w2357_multiples;
typedef struct {
int start_ndx;
int offset;
long int candidate;
heap_t composites;
int count; // count of primes returned.
} primegen_t;
primegen_t make_pgen() {
w2357_multiples *composites;
primegen_t gen;
gen.start_ndx = 0; // primes 2, 3, 5, 7, 11
NEW_PQ_ELE(composites, 121);
gen.offset = composites->offset = 1;
gen.candidate = composites->base_prime = 11;
gen.composites = (heap_t) composites;
gen.count = 0;
return gen;
}
long int next_prime(primegen_t *gen) {
static short upto11[] = {
2, 3, 5, 7, 11
};
if (gen->start_ndx < 5) {
++gen->count;
return upto11[gen->start_ndx++];
} else {
for (;;) {
// advance to the next prime candidate.
gen->candidate += wheel2357[gen->offset++];
if (gen->offset == 48)
gen->offset = 0;
// See if the composite number on top of the heap matches
// the candidate.
//
w2357_multiples *top = (w2357_multiples *) gen->composites;
if (top->hd.key == gen->candidate) { // not prime
do {
// advance the top of heap to the next prime multiple
// that is not a multiple of 2, 3, 5, 7.
//
gen->composites = heap_pop(gen->composites);
top->hd.next = top->hd.down = NULL;
top->hd.key += top->base_prime * wheel2357[top->offset++];
if (top->offset == 48)
top->offset = 0;
gen->composites = heap_merge((heap_t) top, gen->composites);
top = (w2357_multiples *) gen->composites;
} while (top->hd.key == gen->candidate);
} else {
// prime found, add the square and it's position on the wheel
// to the heap.
//
w2357_multiples *new;
HEAP_PUSH(
new,
gen->candidate * gen->candidate,
&gen->composites);
new->offset = gen->offset;
new->base_prime = gen->candidate;
++gen->count;
return gen->candidate;
}
}
}
}
int main() {
primegen_t primes = make_pgen();
printf("first 20: ");
for (int i = 1; i <= 20; i++)
printf("%ld ", next_prime(&primes));
putchar('\n');
printf("between 100 and 150: ");
long int p = next_prime(&primes);
while (p < 150) {
if (p > 100)
printf("%ld ", p);
p = next_prime(&primes);
}
putchar('\n');
int count = 0;
while (p < 8000) {
if (p > 7700)
++count;
p = next_prime(&primes);
}
printf("%d primes between 7700 and 8000.\n", count);
long c;
for (c = 10000; c <= 10000000; c *= 10) {
while (primes.count < c)
p = next_prime(&primes);
printf("%ldth prime is %ld\n", c, p);
}
return 0;
}
```

- Output:

first 20: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 between 100 and 150: 101 103 107 109 113 127 131 137 139 149 30 primes between 7700 and 8000. 10000th prime is 104729 100000th prime is 1299709 1000000th prime is 15485863 10000000th prime is 179424673

## C++

Based on "The Genuine Sieve of Eratosthenes" by Melissa E. O'Neill. UPDATE: Added wheel optimization to match its C counterpart.

```
#include <iostream>
#include <cstdint>
#include <queue>
#include <utility>
#include <vector>
#include <limits>
template<typename integer>
class prime_generator {
public:
integer next_prime();
integer count() const {
return count_;
}
private:
struct queue_item {
queue_item(integer prime, integer multiple, unsigned int wheel_index) :
prime_(prime), multiple_(multiple), wheel_index_(wheel_index) {}
integer prime_;
integer multiple_;
unsigned int wheel_index_;
};
struct cmp {
bool operator()(const queue_item& a, const queue_item& b) const {
return a.multiple_ > b.multiple_;
}
};
static integer wheel_next(unsigned int& index) {
integer offset = wheel_[index];
++index;
if (index == std::size(wheel_))
index = 0;
return offset;
}
typedef std::priority_queue<queue_item, std::vector<queue_item>, cmp> queue;
integer next_ = 11;
integer count_ = 0;
queue queue_;
unsigned int wheel_index_ = 0;
static const unsigned int wheel_[];
static const integer primes_[];
};
template<typename integer>
const unsigned int prime_generator<integer>::wheel_[] = {
2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2,
6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6,
2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10
};
template<typename integer>
const integer prime_generator<integer>::primes_[] = {
2, 3, 5, 7
};
template<typename integer>
integer prime_generator<integer>::next_prime() {
if (count_ < std::size(primes_))
return primes_[count_++];
integer n = next_;
integer prev = 0;
while (!queue_.empty()) {
queue_item item = queue_.top();
if (prev != 0 && prev != item.multiple_)
n += wheel_next(wheel_index_);
if (item.multiple_ > n)
break;
else if (item.multiple_ == n) {
queue_.pop();
queue_item new_item(item);
new_item.multiple_ += new_item.prime_ * wheel_next(new_item.wheel_index_);
queue_.push(new_item);
}
else
throw std::overflow_error("prime_generator: overflow!");
prev = item.multiple_;
}
if (std::numeric_limits<integer>::max()/n > n)
queue_.emplace(n, n * n, wheel_index_);
next_ = n + wheel_next(wheel_index_);
++count_;
return n;
}
int main() {
typedef uint32_t integer;
prime_generator<integer> pgen;
std::cout << "First 20 primes:\n";
for (int i = 0; i < 20; ++i) {
integer p = pgen.next_prime();
if (i != 0)
std::cout << ", ";
std::cout << p;
}
std::cout << "\nPrimes between 100 and 150:\n";
for (int n = 0; ; ) {
integer p = pgen.next_prime();
if (p > 150)
break;
if (p >= 100) {
if (n != 0)
std::cout << ", ";
std::cout << p;
++n;
}
}
int count = 0;
for (;;) {
integer p = pgen.next_prime();
if (p > 8000)
break;
if (p >= 7700)
++count;
}
std::cout << "\nNumber of primes between 7700 and 8000: " << count << '\n';
for (integer n = 10000; n <= 10000000; n *= 10) {
integer prime;
while (pgen.count() != n)
prime = pgen.next_prime();
std::cout << n << "th prime: " << prime << '\n';
}
return 0;
}
```

- Output:

First 20 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 Primes between 100 and 150: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149 Number of primes between 7700 and 8000: 30 10000th prime: 104729 100000th prime: 1299709 1000000th prime: 15485863 10000000th prime: 179424673

### Faster Alternative

This is a "segmented sieve" implementation inspired by the original C solution. Execution time is about 4 seconds on my system (macOS 10.15.4, 3.2GHz Quad-Core Intel Core i5).

```
#include <algorithm>
#include <iostream>
#include <cmath>
#include <cstdint>
#include <vector>
#include <limits>
template<typename integer>
class prime_generator {
public:
explicit prime_generator(integer initial_limit = 100, integer increment = 100000);
integer next_prime();
integer count() const {
return count_;
}
private:
void find_primes(integer);
integer count_ = 0;
integer limit_;
integer index_ = 0;
integer increment_;
std::vector<integer> primes_;
std::vector<bool> sieve_;
integer sieve_limit_ = 0;
};
template<typename integer>
integer next_odd_number(integer n) {
return n % 2 == 0 ? n + 1 : n;
}
template<typename integer>
prime_generator<integer>::prime_generator(integer initial_limit, integer increment)
: limit_(next_odd_number(initial_limit)), increment_(increment) {
primes_.push_back(2);
find_primes(3);
}
template<typename integer>
integer prime_generator<integer>::next_prime() {
if (index_ == primes_.size()) {
if (std::numeric_limits<integer>::max() - increment_ < limit_)
return 0;
int start = limit_ + 2;
limit_ = next_odd_number(limit_ + increment_);
primes_.clear();
find_primes(start);
}
++count_;
return primes_[index_++];
}
template<typename integer>
integer isqrt(integer n) {
return next_odd_number(static_cast<integer>(std::sqrt(n)));
}
template<typename integer>
void prime_generator<integer>::find_primes(integer start) {
index_ = 0;
integer new_limit = isqrt(limit_);
sieve_.resize(new_limit/2);
for (integer p = 3; p * p <= new_limit; p += 2) {
if (sieve_[p/2 - 1])
continue;
integer q = p * std::max(p, next_odd_number((sieve_limit_ + p - 1)/p));
for (; q <= new_limit; q += 2*p)
sieve_[q/2 - 1] = true;
}
sieve_limit_ = new_limit;
size_t count = (limit_ - start)/2 + 1;
std::vector<bool> composite(count, false);
for (integer p = 3; p <= new_limit; p += 2) {
if (sieve_[p/2 - 1])
continue;
integer q = p * std::max(p, next_odd_number((start + p - 1)/p)) - start;
q /= 2;
for (; q < count; q += p)
composite[q] = true;
}
for (integer p = 0; p < count; ++p) {
if (!composite[p])
primes_.push_back(p * 2 + start);
}
}
int main() {
typedef uint64_t integer;
prime_generator<integer> pgen(100, 500000);
std::cout << "First 20 primes:\n";
for (int i = 0; i < 20; ++i) {
integer p = pgen.next_prime();
if (i != 0)
std::cout << ", ";
std::cout << p;
}
std::cout << "\nPrimes between 100 and 150:\n";
for (int n = 0; ; ) {
integer p = pgen.next_prime();
if (p > 150)
break;
if (p >= 100) {
if (n != 0)
std::cout << ", ";
std::cout << p;
++n;
}
}
int count = 0;
for (;;) {
integer p = pgen.next_prime();
if (p > 8000)
break;
if (p >= 7700)
++count;
}
std::cout << "\nNumber of primes between 7700 and 8000: " << count << '\n';
for (integer n = 10000; n <= 100000000; n *= 10) {
integer prime;
while (pgen.count() != n)
prime = pgen.next_prime();
std::cout << n << "th prime: " << prime << '\n';
}
return 0;
}
```

- Output:

First 20 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 Primes between 100 and 150: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149 Number of primes between 7700 and 8000: 30 10000th prime: 104729 100000th prime: 1299709 1000000th prime: 15485863 10000000th prime: 179424673 100000000th prime: 2038074743

## Clojure

```
ns test-project-intellij.core
(:gen-class)
(:require [clojure.string :as string]))
(def primes
" The following routine produces a infinite sequence of primes
(i.e. can be infinite since the evaluation is lazy in that it
only produces values as needed). The method is from clojure primes.clj library
which produces primes based upon O'Neill's paper:
'The Genuine Sieve of Eratosthenes'.
Produces primes based upon trial division on previously found primes up to
(sqrt number), and uses 'wheel' to avoid
testing numbers which are divisors of 2, 3, 5, or 7.
A full explanation of the method is available at:
[https://github.com/stuarthalloway/programming-clojure/pull/12] "
(concat
[2 3 5 7]
(lazy-seq
(let [primes-from ; generates primes by only checking if primes
; numbers which are not divisible by 2, 3, 5, or 7
(fn primes-from [n [f & r]]
(if (some #(zero? (rem n %))
(take-while #(<= (* % %) n) primes))
(recur (+ n f) r)
(lazy-seq (cons n (primes-from (+ n f) r)))))
; wheel provides offsets from previous number to insure we are not landing on a divisor of 2, 3, 5, 7
wheel (cycle [2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2
6 4 6 8 4 2 4 2 4 8 6 4 6 2 4 6
2 6 6 4 2 4 6 2 6 4 2 4 2 10 2 10])]
(primes-from 11 wheel)))))
(defn between [lo hi]
"Primes between lo and hi value "
(->> (take-while #(<= % hi) primes)
(filter #(>= % lo))
))
(println "First twenty:" (take 20 primes))
(println "Between 100 and 150:" (between 100 150))
(println "Number between 7,7700 and 8,000:" (count (between 7700 8000)))
(println "10,000th prime:" (nth primes (dec 10000))) ; decrement by one since nth starts counting from 0
}
```

- Output:

First 20: (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71) Between 100 and 150: (101 103 107 109 113 127 131 137 139 149) Number between 7,700 and 8,000: 30 10000th prime: 104729

**Alternate version using deferred execution Co-Inductive Streams and a Wheel**

The above version is adequate for ranges up to the low millions, so covers the task requirements of primes up to just over a hundred thousands easily. However, it has a O(n^(3/2)) performance which means that it gets slow quite quickly with range as compared to a true incremental Sieve of Eratosthenes, which has O(n (log n)) performance. The following code is about the same speed for ranges in the low millions but quickly passes the above code in speed for large ranges to where it only takes 10's of seconds for a range of a hundred million where the above code takes thousands of seconds. The code is based on the Richard Bird list based Sieve of Eratosthenes mentioned in the O'Neil article but has infinite tree folding added as well as wheel factorization so that it is about the same speed and performance as a Sieve of Eratosthenes based on a priority queue; the code is written here in purely functional form with no mutation, as follows:

```
(deftype CIS [v cont]
clojure.lang.ISeq
(first [_] v)
(next [_] (if (nil? cont) nil (cont)))
(more [this] (let [nv (.next this)] (if (nil? nv) (CIS. nil nil) nv)))
(cons [this o] (clojure.core/cons o this))
(empty [_] (if (and (nil? v) (nil? cont)) nil (CIS. nil nil)))
(equiv [this o] (loop [cis1 this, cis2 o] (if (nil? cis1) (if (nil? cis2) true false)
(if (or (not= (type cis1) (type cis2))
(not= (.v cis1) (.v ^CIS cis2))
(and (nil? (.cont cis1))
(not (nil? (.cont ^CIS cis2))))
(and (nil? (.cont ^CIS cis2))
(not (nil? (.cont cis1))))) false
(if (nil? (.cont cis1)) true
(recur ((.cont cis1)) ((.cont ^CIS cis2))))))))
(count [this] (loop [cis this, cnt 0] (if (or (nil? cis) (nil? (.cont cis))) cnt
(recur ((.cont cis)) (inc cnt)))))
clojure.lang.Seqable
(seq [this] (if (and (nil? v) (nil? cont)) nil this))
clojure.lang.Sequential
Object
(toString [this] (if (and (nil? v) (nil? cont)) "()" (.toString (seq (map identity this))))))
(comment " the wheel could also be a pre-determined vector as for the 2/3/5/7 wheel below...
(def wheel
[ 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2
6 4 6 8 4 2 4 2 4 8 6 4 6 2 4 6
2 6 6 4 2 4 6 2 6 4 2 4 2 10 2 10 ])
")
(def wheel-primes [2 3 5 7 11 13 17])
(def next-prime 19)
(def nextnext-prime 23)
;; calculates the vector for very large wheels such as the 92160 element version here
;; the disadvantage is that it takes some time to calculate before the work can start...
(def wheel
(loop [p 2, len 1, ^bytes ptrn [1]]
(if (>= p next-prime)
ptrn
(let [cptrn (cycle ptrn), [f & rcyc] cptrn,
np (+ p f), nlen (* len (- p 1)),
culls
(map (fn [[f _]] f)
(iterate (fn [[c [g & r]]] [(+ c (* p g)) r]) [(* p p) cptrn])),
gaps (drop 1
(for [[gp _ _ _ cnt]
(iterate (fn [[_ v cls [g & rgs] c]]
(let [[cl & rcls] cls, tv (+ v g),
[sg & srgs] rgs, nc (+ c 1)]
(if (= cl tv)
[(+ g sg) (+ tv sg) rcls srgs nc]
[g tv cls rgs nc])))
[f np culls rcyc 0]) :while (<= cnt nlen)] gp))]
(recur np nlen (vec gaps))))))
(def wheellmt (- (count wheel) 1))
(defn primes-treeFolding
"Computes the unbounded sequence of primes using a Sieve of Eratosthenes algorithm modified from Bird."
[]
(letfn [(mltpls [[p pi]]
(letfn [(nxtmltpl [c ci]
(let [nci (if (< ci wheellmt) (+ ci 1) 0)]
(->CIS c #(-> (nxtmltpl (+ c (* p (get wheel ci))) nci)))))]
(nxtmltpl (* p p) pi))),
(allmtpls [^CIS pxs]
(->CIS (mltpls (.v pxs)) #(-> (allmtpls ((.cont pxs)))))),
(union [^CIS xs ^CIS ys]
(let [xv (.v xs), yv (.v ys)]
(if (< xv yv) (->CIS xv #(-> (union ((.cont xs)) ys)))
(if (< yv xv)
(->CIS yv #(-> (union xs ((.cont ys)))))
(->CIS xv #(-> (union (next xs) ((.cont ys))))))))),
(pairs [^CIS mltplss] (let [^CIS tl ((.cont mltplss))]
(->CIS (union (.v mltplss) (.v tl))
#(-> (pairs ((.cont tl))))))),
(mrgmltpls [^CIS mltplss]
(->CIS (.v ^CIS (.v mltplss))
#(-> (union ((.cont ^CIS (.v mltplss)))
(mrgmltpls (pairs ((.cont mltplss)))))))),
(minusStrtAt [n ni ^CIS cmpsts]
(let [nn (+ n (get wheel ni)), nni (if (< ni wheellmt) (+ ni 1) 0)]
(if (< n (.v cmpsts))
(->CIS [n ni] #(-> (minusStrtAt nn nni cmpsts)))
(recur nn nni ((.cont cmpsts)))))),
(xtraprmsndxd []
(->CIS [next-prime 0] #(-> (minusStrtAt nextnext-prime 1
(mrgmltpls (allmtpls (xtraprmsndxd))))))),
(stripndxs [^CIS ndxd]
(->CIS (get (.v ndxd) 0) #(-> (stripndxs ((.cont ndxd))))))]
(loop [i (- (count wheel-primes) 1), ff (fn [] (stripndxs (xtraprmsndxd)))]
(if (<= i 0)
(->CIS (get wheel-primes 0) ff)
(recur (- i 1) (fn [] (->CIS (get wheel-primes i) ff)))))))
```

Now these functional incremental sieves are of limited use if one requires ranges of billions as they are hundreds of times slower than a version of a bit-packed page-segmented mutable array Sieve of Eratosthenes, which for Clojure there is a version at the end of Clojure Sieve_of_Eratosthenes#Unbounded_Versions section on the Sieve of Eratosthenes task page; this version will handle ranges of a billion in seconds rather than hundreds of seconds.

## CoffeeScript

This uses the prime number generation algorithm outlined in the paper, "Two Compact Incremental Prime Sieves" by Jonathon P. Sorenson. This algorithm is essentially a rolling segmented SoE, and is quite fast for languages that have good array processing.

```
primes = () ->
yield 2
yield 3
sieve = ([] for i in [1..3])
sieve[0].push 3
[r, s] = [3, 9]
pos = 1
n = 5
loop
isPrime = true
if sieve[pos].length > 0 # this entry has a list of factors
isPrime = false
sieve[(pos + m) % sieve.length].push m for m in sieve[pos]
sieve[pos] = []
if n is s # n is the next square
if isPrime
isPrime = false # r divides n, so not actually prime
sieve[(pos + r) % sieve.length].push r # however, r is prime
r += 2
s = r*r
yield n if isPrime
n += 2
pos += 1
if pos is sieve.length
sieve.push [] # array size must exceed largest prime found
sieve.push [] # adding two entries keeps size = O(sqrt n)
pos = 0
undefined # prevent CoffeeScript from aggregating values
module.exports = {
primes
}
```

Driver code:

```
primes = require('sieve').primes
gen = primes()
console.log "The first 20 primes: #{gen.next().value for _ in [1..20]}"
p100_150 = (while (p = gen.next().value) < 150 then p).filter (n) -> n > 100
console.log "The primes between 100 and 150: #{p100_150}"
while gen.next().value < 7700
undefined
count = 1
while gen.next().value < 8000
++count
console.log "There are #{count} primes between 7,700 and 8,000."
n = 10
c = 0
gen = primes()
loop
p = gen.next().value
c += 1
if c is n
console.log "The #{n}th prime is #{p}"
break if n is 10_000_000
n *= 10
```

- Output:

The first 20 primes: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71 The primes between 100 and 150: 101,103,107,109,113,127,131,137,139,149 There are 30 primes between 7,700 and 8,000. The 10th prime is 29 The 100th prime is 541 The 1000th prime is 7919 The 10000th prime is 104729 The 100000th prime is 1299709 The 1000000th prime is 15485863

## D

This uses a Prime struct defined in the third entry of the Sieve of Eratosthenes task. Prime keeps and extends a dynamic array instance member of uints. The Prime struct has a opCall that returns the n-th prime number. The opCall calls a grow() private method until the dynamic array of primes is long enough to contain the required answer. The function grow() just grows the dynamic array geometrically and performs a normal sieving. On a 64 bit system this program works up to the maximum prime number that can be represented in the 32 bits of an uint. This program is less efficient than the C entry, so it's better to not use it past some tens of millions of primes, but it's enough for more limited usages.

```
void main() {
import std.stdio, std.range, std.algorithm, sieve_of_eratosthenes3;
Prime prime;
writeln("First twenty primes:\n", 20.iota.map!prime);
writeln("Primes primes between 100 and 150:\n",
uint.max.iota.map!prime.until!q{a > 150}.filter!q{a > 99});
writeln("Number of primes between 7,700 and 8,000: ",
uint.max.iota.map!prime.until!q{a > 8_000}
.filter!q{a > 7_699}.walkLength);
writeln("10,000th prime: ", prime(9_999));
}
```

- Output:

First twenty primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71] Primes primes between 100 and 150: [101, 103, 107, 109, 113, 127, 131, 137, 139, 149] Number of primes between 7,700 and 8,000: 30 10,000th prime: 104729

### Faster Alternative Version

```
/// Prime sieve based on: http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf
import std.container: Array, BinaryHeap, RedBlackTree;
struct LazyPrimeSieve {
@property bool empty() const pure nothrow @safe @nogc {
return i > 203_280_221; // Pi(2 ^^ 32).
}
@property auto front() const pure nothrow @safe @nogc {
return prime;
}
@property void popFront() pure nothrow /*@safe*/ {
prime = sieveOne();
}
private:
static struct Wheel2357 {
static immutable ubyte[48] holes = [2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6,
2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6,
4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10];
static immutable ubyte[4] spokes = [2, 3, 5, 7];
static immutable ubyte first = 11;
uint i;
auto spin() pure nothrow @safe @nogc {
return holes[i++ % $];
}
}
static struct CompositeIterator {
uint prime;
Wheel2357 wheel;
ulong composite;
this(uint p) pure nothrow @safe @nogc {
prime = p;
composite = p * wheel.first;
}
void next() pure nothrow @safe @nogc {
composite += prime * wheel.spin;
}
}
version (heap) // Less memory but slower.
BinaryHeap!(Array!CompositeIterator, "a.composite > b.composite") iterators;
else // Faster but is more GC intensive.
RedBlackTree!(CompositeIterator, "a.composite < b.composite", true) iterators;
uint prime = 2;
uint i = 1;
Wheel2357 wheel;
uint candidate = wheel.first;
uint sieveOne() pure nothrow /*@safe*/ {
switch (i) {
case 0: .. case wheel.spokes.length - 1:
return wheel.spokes[i++];
case wheel.spokes.length:
i++;
return candidate;
case wheel.spokes.length + 1:
version (heap) {}
else
iterators = new typeof(iterators);
goto default;
default:
goto POST_RETURN;
while (true) {
candidate += wheel.spin;
while (iterators.front.composite < candidate) {
auto it = iterators.front;
iterators.removeFront;
it.next;
iterators.insert(it);
}
if (iterators.front.composite != candidate) {
i++;
return candidate;
POST_RETURN:
// Only insert primes that are multiply
// occuring in [0, 2 ^^ 32).
if (candidate < 2 ^^ 16)
iterators.insert(CompositeIterator(candidate));
}
}
}
}
}
void main() /*@safe*/ {
import std.stdio, std.algorithm, std.range;
writeln("Sum of first 100,000 primes: ", LazyPrimeSieve().take(100_000).sum(0uL));
writeln("First twenty primes:\n", LazyPrimeSieve().take(20));
writeln("Primes primes between 100 and 150:\n",
LazyPrimeSieve().until!q{a > 150}.filter!q{a > 99});
writeln("Number of primes between 7,700 and 8,000: ",
LazyPrimeSieve().until!q{a > 8_000}.filter!q{a > 7_699}.walkLength);
writeln("10,000th prime: ", LazyPrimeSieve().dropExactly(9999).front);
}
```

- Output:

Sum of first 100,000 primes: 62260698721 First twenty primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71] Primes primes between 100 and 150: [101, 103, 107, 109, 113, 127, 131, 137, 139, 149] Number of primes between 7,700 and 8,000: 30 10,000th prime: 104729

## Dart

A version based on a (hashed) Map:

```
Iterable<int> primesMap() {
Iterable<int> oddprms() sync* {
yield(3); yield(5); // need at least 2 for initialization
final Map<int, int> bpmap = {9: 6};
final Iterator<int> bps = oddprms().iterator;
bps.moveNext(); bps.moveNext(); // skip past 3 to 5
int bp = bps.current;
int n = bp;
int q = bp * bp;
while (true) {
n += 2;
while (n >= q || bpmap.containsKey(n)) {
if (n >= q) {
final int inc = bp << 1;
bpmap[bp * bp + inc] = inc;
bps.moveNext(); bp = bps.current; q = bp * bp;
} else {
final int inc = bpmap.remove(n);
int next = n + inc;
while (bpmap.containsKey(next)) {
next += inc;
}
bpmap[next] = inc;
}
n += 2;
}
yield(n);
}
}
return [2].followedBy(oddprms());
}
void main() {
print("The first 20 primes:");
String str = "( ";
primesMap().take(20).forEach((p)=>str += "$p "); print(str + ")");
print("Primes between 100 and 150:");
str = "( ";
primesMap().skipWhile((p)=>p<100).takeWhile((p)=>p<150)
.forEach((p)=>str += "$p "); print(str + ")");
print("Number of primes between 7700 and 8000: ${
primesMap().skipWhile((p)=>p<7700).takeWhile((p)=>p<8000).length
}");
print("The 10,000th prime: ${
primesMap().skip(9999).first
}");
final start = DateTime.now().millisecondsSinceEpoch;
final answer = primesMap().takeWhile((p)=>p<2000000).reduce((a,p)=>a+p);
final elapsed = DateTime.now().millisecondsSinceEpoch - start;
}
```

- Output:

The first 20 primes: ( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ) Primes between 100 and 150: ( 101 103 107 109 113 127 131 137 139 149 ) Number of primes between 7700 and 8000: 30 The 10,000th prime: 104729 The sum of the primes to two million: 142913828922 This test bench took 356 milliseconds.

This version has a O(n log (log n)) computational complexity due to hash map access being O(1) on average but is somewhat slow due to the constant execution overhead and therefore only somewhat useful for ranges of up to about the tens of millions.

As a bonus, it solves Euler Problem 10 of summing all the primes up to two million quite quickly.

**A faster alternative version based on the infinite page segmented sieve**

The unbounded page segmented bit-packed version from the Sieve_of_Eratosthenes#Unbounded_infinite_iterators.2Fgenerators_of_primes at the bottom of the section can do the same job tens of times faster just by substituting 'primesPaged()' for 'primesMap()' in the 'main' function above in all places used. As noted for the listing on the task page, the code is only limited in range by the integer size limit on 32-bit execution environments, and will be tens of times faster than the above version. It will otherwise have the same output as above.

It solves the Euler Problem 10 in almost too short a time to be measured, and it becomes useful for ranges of hundreds of thousands. It can count all the primes to a billion on the low end tablet CPU of an Intel x5-Z8350 at 1.92 Gigahertz used to develop this in 40 seconds but using a generator slows the performance and it can use the provided `countPrimesTo` function to do the job four times as fast by directly manipulating the provided iteration of sieved bit-packed arrays.

## Delphi

This prime generator is based on a custom Delphi object. The object has a number of features:

1. It sieves primes generating an array of flags that tells whether a number is prime or not.

2. Using that table it generates another table of just prime numbers.

3. The tables can be up 4 billion flags under a 32-bit operating system. However there is also the option to use "Bit-Booleans" which can push the limit up to 32

4. Tables are generated fast; 1 million primes takes 9 miliseconds to generate; 100 million primes requires about 2 seconds.

5. The tool can be used from just about any prime based task. You can determine if a number is prime by testing the corresponding flag. You can find the nth prime by looking at the nth entry int he prime table. You can count primes between two values by counting flags.

```
{{-------- Declaration for BitBoolean Array ------------------}
{Bit boolean - because it stores 8 bools per bytye, it will}
{handle up to 16 gigabyte in a 32 bit programming environment}
type TBitBoolArray = class(TObject)
private
FSize: int64;
ByteArray: array of Byte;
function GetValue(Index: int64): boolean;
procedure WriteValue(Index: int64; const Value: boolean);
function GetSize: int64;
procedure SetSize(const Value: int64);
protected
public
property Value[Index: int64]: boolean read GetValue write WriteValue; default;
constructor Create;
property Count: int64 read GetSize write SetSize;
procedure Clear(Value: boolean);
end;
{------------------------------------------------------------}
{ Implementation for Bitboolean array -----------------------}
{------------------------------------------------------------}
{ TBitBoolArray }
const BitArray: array [0..7] of byte = ($01, $02, $04, $08, $10, $20, $40, $80);
function TBitBoolArray.GetValue(Index: int64): boolean;
begin
{Note: (Index and 7) is faster than (Index mod 8)}
Result:=(ByteArray[Index shr 3] and BitArray[Index and 7])<>0;
end;
procedure TBitBoolArray.WriteValue(Index: int64; const Value: boolean);
var Inx: int64;
begin
Inx:=Index shr 3;
{Note: (Index and 7) is faster than (Index mod 8)}
if Value then ByteArray[Inx]:=ByteArray[Inx] or BitArray[Index and 7]
else ByteArray[Inx]:=ByteArray[Inx] and not BitArray[Index and 7]
end;
constructor TBitBoolArray.Create;
begin
SetLength(ByteArray,0);
end;
function TBitBoolArray.GetSize: int64;
begin
Result:=FSize;
end;
procedure TBitBoolArray.SetSize(const Value: int64);
var Len: int64;
begin
FSize:=Value;
{Storing 8 items per byte}
Len:=Value div 8;
{We need one more to fill partial bits}
if (Value mod 8)<>0 then Inc(Len);
SetLength(ByteArray,Len);
end;
procedure TBitBoolArray.Clear(Value: boolean);
var Fill: byte;
begin
if Value then Fill:=$FF else Fill:=0;
FillChar(ByteArray[0],Length(ByteArray),Fill);
end;
{========== TPrimeSieve =======================================================}
{Sieve object the generates and holds prime values}
{Enable this flag if you need primes past 2 billion.
The flag signals the code to use bit-booleans arrays
which can contain up to 8 x 4 gigabytes = 32 gig booleans.}
// {$define BITBOOL}
type TPrimeSieve = class(TObject)
private
{$ifdef BITBOOL}
PrimeArray: TBitBoolArray;
{$else}
PrimeArray: array of boolean;
{$endif}
FArraySize: int64;
FPrimeCount: int64;
function GetPrime(Index: int64): boolean;
procedure Clear;
function GetCount: int64;
procedure BuildPrimeTable;
protected
procedure DoSieve;
property ArraySize: int64 read FArraySize;
public
Primes: TIntegerDynArray;
BitBoolean: boolean;
constructor Create;
destructor Destroy; override;
procedure Intialize(Size: int64);
property Flags[Index: int64]: boolean read GetPrime; default;
function NextPrime(Start: int64): int64;
function PreviousPrime(Start: int64): int64;
property Count: int64 read GetCount;
property PrimeCount: int64 read FPrimeCount;
end;
procedure TPrimeSieve.Clear;
begin
{$ifdef BITBOOL}
PrimeArray.Clear(True);
{$else}
FillChar(PrimeArray[0],Length(PrimeArray),True);
{$endif}
end;
constructor TPrimeSieve.Create;
begin
{$ifdef BITBOOL}
PrimeArray:=TBitBoolArray.Create;
BitBoolean:=True;
{$else}
BitBoolean:=False;
{$endif}
end;
destructor TPrimeSieve.Destroy;
begin
{$ifdef BITBOOL}
PrimeArray.Free;
{$endif}
inherited;
end;
procedure TPrimeSieve.BuildPrimeTable;
{This builds a table of primes which is}
{easier to use than a table of flags}
var I,Inx: integer;
begin
SetLength(Primes,Self.PrimeCount);
Inx:=0;
for I:=0 to Self.Count-1 do
if Flags[I] then
begin
Primes[Inx]:=I;
Inc(Inx);
end;
end;
procedure TPrimeSieve.DoSieve;
{Load flags with true/false to flag that number is prime}
{Note: does not store even values, because except for 2, all primes are even}
{Starts storing flags at Index=3, so reading/writing routines compensate}
{Uses for-loops for boolean arrays and while-loops for Bit-Booleans arrays}
{$ifdef BITBOOL}
var Offset, I, K: int64;
{$else}
var Offset, I, K: cardinal;
{$endif}
begin
Clear;
{Compensate from primes 1,2 & 3, which aren't stored}
FPrimeCount:=ArraySize+3;
{$ifdef BITBOOL}
I:=0;
while I<ArraySize do
{$else}
for I:=0 to ArraySize-1 do
{$endif}
begin
if PrimeArray[I] then
begin
Offset:= I + I + 3;
K:= I + Offset;
while K <=(ArraySize-1) do
begin
if PrimeArray[K] then Dec(FPrimeCount);
PrimeArray[K]:= False;
K:= K + Offset;
end;
end;
{$ifdef BITBOOL} Inc(I); {$endif}
end;
BuildPrimeTable;
end;
function TPrimeSieve.GetPrime(Index: int64): boolean;
{Get a prime flag from array - compensates}
{ for 0,1,2 and even numbers not being stored}
begin
if Index = 1 then Result:=False
else if Index = 2 then Result:=True
else if (Index and 1)=0 then Result:=false
else Result:=PrimeArray[(Index div 2)-1];
end;
function TPrimeSieve.NextPrime(Start: int64): int64;
{Get next prime after Start}
begin
Result:=Start+1;
while Result<=((ArraySize-1) * 2) do
begin
if Self.Flags[Result] then break;
Inc(Result);
end;
end;
function TPrimeSieve.PreviousPrime(Start: int64): int64;
{Get Previous prime Before Start}
begin
Result:=Start-1;
while Result>0 do
begin
if Self.Flags[Result] then break;
Dec(Result);
end;
end;
procedure TPrimeSieve.Intialize(Size: int64);
{Set array size and do Sieve to load flag array with}
begin
FArraySize:=Size div 2;
{$ifdef BITBOOL}
PrimeArray.Count:=FArraySize;
{$else}
SetLength(PrimeArray,FArraySize);
{$endif}
DoSieve;
end;
function TPrimeSieve.GetCount: int64;
begin
Result:=FArraySize * 2;
end;
{===========================================================}
procedure ExtensiblePrimeGenerator(Memo: TMemo);
var I,Cnt: integer;
var Sieve: TPrimeSieve;
var S: string;
begin
Sieve:=TPrimeSieve.Create;
try
{Build a table with 1-million primes}
Sieve.Intialize(1000000);
Memo.Lines.Add('Showing the first twenty primes');
S:='';
for I:=0 to 20-1 do
S:=S+' '+IntToStr(Sieve.Primes[I]);
Memo.Lines.Add(S);
Memo.Lines.Add('');
Memo.Lines.Add('Showing the primes between 100 and 150.');
S:='';
for I:=100 to 150 do
if Sieve.Flags[I] then S:=S+' '+IntToStr(I);
Memo.Lines.Add(S);
Memo.Lines.Add('');
Memo.Lines.Add('Showing the number of primes between 7,700 and 8,000.');
Cnt:=0;
for I:=7700 to 8000 do
if Sieve.Flags[I] then Inc(Cnt);
Memo.Lines.Add('Count = '+IntToStr(Cnt));
Memo.Lines.Add('');
Memo.Lines.Add('Showing the 10,000th prime.');
Memo.Lines.Add('10,000th Prime = '+IntToStr(Sieve.Primes[10000-1]));
finally Sieve.Free; end;
end;
```

- Output:

Showing the first twenty primes 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 Showing the primes between 100 and 150. 101 103 107 109 113 127 131 137 139 149 Showing the number of primes between 7,700 and 8,000. Count = 30 Showing the 10,000th prime. 10,000th Prime = 104729 Elapsed Time: 19.853 ms.

## EasyLang

```
fastfunc nprim num .
repeat
i = 2
while i <= sqrt num and num mod i <> 0
i += 1
.
until num mod i <> 0
num += 1
.
return num
.
prim = 2
primcnt = 1
proc nextprim . .
prim = nprim (prim + 1)
primcnt += 1
.
for i to 20
write prim & " "
nextprim
.
print ""
while prim < 100
nextprim
.
while prim <= 150
write prim & " "
nextprim
.
print ""
while prim < 7700
nextprim
.
while prim <= 8000
cnt += 1
nextprim
.
print cnt
while primcnt < 10000
nextprim
.
print prim
while primcnt < 100000
nextprim
.
print prim
```

## EchoLisp

Standard prime functions handle numbers < 2e+9. See [1] . The **bigint** library handles large numbers. See [2]. The only limitations are time, memory, and browser performances ..

```
; the first twenty primes
(primes 20)
→ { 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 }
; a stream to generate primes from a
(define (primes-from a)
(let ((p (next-prime a)))
(stream-cons p (primes-from p))))
; primes between 100,150
(for/list ((p (primes-from 100))) #:break (> p 150) p)
→ (101 103 107 109 113 127 131 137 139 149)
; the built-in function (primes-pi )counts the number of primes < a
; count in [7700 ... 8000]
(- (primes-pi 8000) (primes-pi 7700) → 30
; nth-prime
(nth-prime 10000) → 104729
;; big ones
(lib 'bigint)
(define (p-digits n)
(printf "(next-prime %d ! ) has %d digits" n
(number-length (next-prime (factorial n )))))
(next-prime 0! ) has 1 digits
(next-prime 10! ) has 7 digits
(next-prime 100! ) has 158 digits
(next-prime 200! ) has 375 digits
(next-prime 300! ) has 615 digits
(next-prime 400! ) has 869 digits ;; 9400 msec (FireFox)
; is prime (1 + 116!) ?
(prime? (1+ (factorial 116))) → #t
```

## Elixir

The Sieve_of_Eratosthenes#Elixir Task page lists two "infinite" extensible generators at the bottom. The first of those, using a (hash) Map is reproduced here along with the code to fulfill the required tasks:

```
defmodule PrimesSoEMap do
@typep stt :: {integer, integer, integer, Enumerable.integer, %{integer => integer}}
@spec advance(stt) :: stt
defp advance {n, bp, q, bps?, map} do
bps = if bps? === nil do Stream.drop(oddprms(), 1) else bps? end
nn = n + 2
if nn >= q do
inc = bp + bp
nbps = bps |> Stream.drop(1)
[nbp] = nbps |> Enum.take(1)
advance {nn, nbp, nbp * nbp, nbps, map |> Map.put(nn + inc, inc)}
else if Map.has_key?(map, nn) do
{inc, rmap} = Map.pop(map, nn)
[next] =
Stream.iterate(nn + inc, &(&1 + inc))
|> Stream.drop_while(&(Map.has_key?(rmap, &1))) |> Enum.take(1)
advance {nn, bp, q, bps, Map.put(rmap, next, inc)}
else
{nn, bp, q, bps, map}
end end
end
@spec oddprms() :: Enumerable.integer
defp oddprms do # put first base prime cull seq in Map so never empty
# advance base odd primes to 5 when initialized
init = {7, 5, 25, nil, %{9 => 6}}
[3, 5] # to avoid race, preseed with the first 2 elements...
|> Stream.concat(
Stream.iterate(init, &(advance &1))
|> Stream.map(fn {p,_,_,_,_} -> p end))
end
@spec primes() :: Enumerable.integer
def primes do
Stream.concat([2], oddprms())
end
end
IO.write "The first 20 primes are:\n( "
PrimesSoEMap.primes() |> Stream.take(20) |> Enum.each(&(IO.write "#{&1} "))
IO.puts ")"
IO.write "The primes between 100 to 150 are:\n( "
PrimesSoEMap.primes() |> Stream.drop_while(&(&1<100))
|> Stream.take_while(&(&1<150)) |> Enum.each(&(IO.write "#{&1} "))
IO.puts ")"
IO.write "The number of primes between 7700 and 8000 is: "
PrimesSoEMap.primes() |> Stream.drop_while(&(&1<7700))
|> Stream.take_while(&(&1<8000)) |> Enum.count |> IO.puts
IO.write "The 10,000th prime is: "
PrimesSoEMap.primes() |> Stream.drop(9999)
|> Enum.take(1) |> List.first |>IO.puts
IO.write "The sum of all the priems to two million is: "
testfunc =
fn () ->
ans =
PrimesSoEMap.primes() |> Stream.take_while(&(&1<=2000000))
|> Enum.sum() |> IO.puts
ans end
:timer.tc(testfunc)
|> (fn {t,_} ->
IO.puts "This test bench took #{t} microseconds." end).()
```

- Output:

The first 20 primes are: ( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ) The primes between 100 to 150 are: ( 101 103 107 109 113 127 131 137 139 149 ) The number of primes between 7700 and 8000 is: 30 The 10,000th prime is: 104729 The sum of all the primes to two million is: 142913828922 This test bench took 7827912 microseconds.

The code solves the (trivial) task requirements quickly, but being purely functional, running on a Virtual Machine, and using the multi-precision ("Big Integer") `integer` type is somewhat slower than as implemented in some other languages.

The code is an "infinite" generator as the `integer` type in Elixir is of multi-precision and thus will never run out of range.

As a bonus, the above code solves the Euler Problem 10 of summing the primes to two million in about 7.83 seconds, or about a hundred thousand CPU cycles per prime on a !.92 Gigahertz CPU, which at least is within the 30 second time limit for that problem.

**Alternate somewhat faster version (over two times)**

The last code on the Sieve of Eratosthenes Task page uses deferred execution Co-Inductive Streams (CIS's) to implement the incremental functional Sieve of Eratosthenes using an infinite tree folding structure with a final output as a lazy Stream just as for the above. It is about twice as fast as the above for reasonable ranges of a few millions. It con be used just by substituting calls to the different named module as in `PrimesSoETreeFolding.primes` rather than `PrimesSoEMap.primes()`. It has the same limitations as to being "infinite" as the above.

## F#

This task uses Unbounded_Page-Segmented_Bit-Packed Odds-Only Mutable Array Sieve F#

### The functions

```
let primeZ fN =primes()|>Seq.unfold(fun g-> Some(fN(g()), g))
let primesI() =primeZ bigint
let primes64() =primeZ int64
let primes32() =primeZ int32
let pCache =Seq.cache(primes32())
let isPrime g=if g<2 then false else let mx=int(sqrt(float g)) in pCache|>Seq.takeWhile(fun n->n<=mx)|>Seq.forall(fun n->g%n>0)
let isPrime64 g=if g<2L then false else let mx=int(sqrt(float g)) in pCache|>Seq.takeWhile(fun n->n<=mx)|>Seq.forall(fun n->g%(int64 n)>0L)
```

### The Task

```
Seq.take 20 primes32()|> Seq.iter (fun n-> printf "%d " n)
```

- Output:

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

```
primes32() |> Seq.skipWhile (fun n->n<100) |> Seq.takeWhile (fun n->n<=150) |> Seq.iter (fun n -> printf "%d " n)
```

- Output:

101 103 107 109 113 127 131 137 139 149

```
printfn "%d" (primes32() |> Seq.skipWhile (fun n->n<7700) |> Seq.takeWhile (fun n->n<=8000) |> Seq.length)
```

- Output:

30

To demonstrate extensibility I find the 10000th prime.

```
Seq.item 9999 pCache
```

- Output:

Real: 00:00:00.185, CPU: 00:00:00.190, GC gen0: 21, gen1: 0 val it : int = 104729

I then find the 10001st prime which takes less time.

```
Seq.item 10000 pCache
```

- Output:

Real: 00:00:00.004, CPU: 00:00:00.010, GC gen0: 1, gen1: 0 val it : int = 104743

To indiccate speed I time the following:

```
let strt = System.DateTime.Now.Ticks
for i = 1 to 8 do
let n = pown 10 i // the item index below is zero based!
printfn "The %dth prime is: %A" n (primeZ int |> Seq.item (n - 1))
let timed = (System.DateTime.Now.Ticks - strt) / 10000L
printfn "All of the last took %d milliseconds." timed
```

- Output:

The 10th prime is: 29 The 100th prime is: 541 The 1000th prime is: 7919 The 10000th prime is: 104729 The 100000th prime is: 1299709 The 1000000th prime is: 15485863 The 10000000th prime is: 179424673 The 100000000th prime is: 2038074743 All of the last took 7937 milliseconds.

```
printfn "The first 20 primes are: %s"
( primesSeq() |> Seq.take 20
|> Seq.fold (fun s p -> s + string p + " ") "" )
printfn "The primes from 100 to 150 are: %s"
( primesSeq() |> Seq.skipWhile ((>) (prime 100))
|> Seq.takeWhile ((>=) (prime 150))
|> Seq.fold (fun s p -> s + string p + " ") "" )
printfn "The number of primes from 7700 to 8000 are: %d"
( primesSeq() |> Seq.skipWhile ((>) (prime 7700))
|> Seq.takeWhile ((>=) (prime 8000)) |> Seq.length )
let strt = System.DateTime.Now.Ticks
for i = 1 to 8 do
let n = pown 10 i // the item index below is zero based!
printfn "The %dth prime is: %A" n (primesSeq() |> Seq.item (n - 1))
let timed = (System.DateTime.Now.Ticks - strt) / 10000L
printfn "All of the last took %d milliseconds." timed
```

- Output:

The first 20 primes are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 The primes from 150 to 150 are: 101 103 107 109 113 127 131 137 139 149 The number of primes from 7700 to 8000 are: 30 The 10th prime is: 29 The 100th prime is: 541 The 1000th prime is: 7919 The 10000th prime is: 104729 The 100000th prime is: 1299709 The 1000000th prime is: 15485863 The 10000000th prime is: 179424673 The 100000000th prime is: 2038074743 All of the last took 16634 milliseconds.

Even at that, this is slow due to the time to enumerate and there are much faster ways to do this without using enumeration as in manipulating the found prime bit representations directly...

## Factor

Factor's math.primes vocabulary provides an extensible primality test upon which its prime number generators are built. For values below nine million, it checks for primality using a Sieve of Eratosthenes with wheels. For values nine million and above, it uses a Miller-Rabin probabalistic primality test.

Factor's `fixnums`

automatically promote to `bignums`

when large enough, so there are no limits to its prime generator other than the capabilities of the machine it's running on.

```
USING: io math.primes prettyprint sequences ;
"First 20 primes: " write
20 nprimes .
"Primes between 100 and 150: " write
100 150 primes-between .
"Number of primes between 7,700 and 8,000: " write
7,700 8,000 primes-between length .
"10,000th prime: " write
10,000 nprimes last .
```

- Output:

First 20 primes: { 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 } Primes between 100 and 150: V{ 101 103 107 109 113 127 131 137 139 149 } Number of primes between 7,700 and 8,000: 30 10,000th prime: 104729

## Fortran

### The Plan

Over the years, the storage of boolean variables has been a steady source of vexation. Few systems offer operation codes that can work on individual bits, so the usual approach is to allocate a convenient storage unit to hold the value. In Fortran, the default size of a LOGICAL variable is the same as that of an INTEGER variable, and these days, that means thirty-two bits to store the state of one. However, with the increasing use of character manipulation rather than just numbers, there is often support in the cpu for single-character access and some later Fortran compilers will recognise LOGICAL*1 and so reserve only eight bits per boolean variable. Even so, any attempt to store a boolean variable in a single bit will for every access require code to isolate that bit from the rest of the storage unit where it resides, and these operation codes will require more storage than would be saved. Similarly with collections of variables: some might best be aligned to word boundaries (and for double-sized variables, perhaps to even word boundaries) so the storage plan may well involve adding "padding" to preserve such alignment. Some languages (such as pl/i) offer a word ALIGNED to ensure this, and others (such as Pascal) offer PACKED for cramming, of use when dealing with records for a disc file and intending to save space. So, ... if there is a large array of boolean variables, and, there are not so many references to those variables, there is still an opportunity.

And indeed, the array shall be large. Some simple investigations show that storing a collection of prime numbers in an array of integers occupies rather more storage than storing a simple bit array spanning the same range of numbers, and given the obvious scheme of storing bits only for odd integers, this advantage is still greater - see the schedule in the source file. One could argue that the array of successive prime numbers could be stored in various space-saving ways, but, so also can the bit array be compressed. For instance, have a span of a "primorial" size such as 2*3*5 and a reference span with those factors marked "off": that removes seven odd numbers from consideration, leaving eight candidates for each surge and so only eight bits are required to state "prime" or not instead of fifteen. With non-binary computers, "bit fiddling" is less convenient but still possible. One must use techniques similar to those needed to work with the year, month, and day parts of an integer such as 20161015 in binary.

So, the plan is to have a long array of bits, and, rather than commit a lot of memory to this, do so in a disc file with random access. To follow the "extensible" aspect, this disc file will *not* be initialised to its maximum extent on the first invocation of the routine, instead, it will be extended as provoked by requests for NEXTPRIME and so forth.

#### Initialisation

When arranging a sieve of Eratosthenes, one of the problems is that one wishes to step along only with steps of prime number size to avoid wasted effort, but, before the sieve process is completed, there is no ready source of known prime numbers. This is especially difficult when instead of one long sieve covering the whole span of interest, the process is to proceed in surges, repeatedly using some limited size span. For this reason, it is often convenient to prepare an initial array of prime numbers knowing for example that Prime(4792) = 46337, and that the square of the next prime exceeds the range of signed 32-bit integers. But such pre-emptive preparation conflicts with the "extensible" notion, and requires special code and storage for the array.

Because Fortran passes parameters by reference (i.e. by address of the original) a trick is possible. The array SCHARS is shared storage to hold a record from the disc file (as a "buffer") and when subroutine GRASPPRIMEBAG is invoked to gain access to its disc file it notes whether it must create the file. If so, the first record is to be written, and the call is PSURGE(SCHARS) to do so within the shared bitpad. PSURGE knows that its first stepper is with `F = 3`

(because even numbers are not being represented) and proceeds with that, adjusting array SCHARS. When it is ready for the next sieve pass, it invokes `F = NEXTPRIME(F)`

to find the next stepper, which will be five, and NEXTPRIME scans the bit array in SCHARS to find it. *This is the same bitpad that PSURGE is in the process of adjusting.* To support this startup ploy, GETSREC (invoked by NEXTPRIME) returns at once when SLAST = 0, signifying that there are no records in the work file as yet. Later, if GETSREC determines that the bit array is to be extended, it invokes PSURGE with its local array BIT8 as the bitpad to be developed then written to disc, leaving the shared SCHAR array as a record buffer for the use of NEXTPRIME when invoked by PSURGE.

#### Supporting NextPrime(n)

Since the bit array has a simple linear relationship to the numbers it is associated with, function `NEXTPRIME(n)`

(and `PREVIOUSPRIME(n)`

) can easily calculate the index to access the appropriate bits; similarly, function `ISPRIME(n)`

need merely check `n = NEXTPRIME(n - 1)`

rather than slog through possibly all potential prime factors up to SQRT(n) - though this does mean that prime numbers up to *n* must be available rather than merely up to SQRT(n). [A later adjustment has ISPRIME(n) repeat the code to locate the bit for *n* rather than use NEXTPRIME, which has to scan the bit array to the next prime] But there would be no escape from such a slog for function FIRSTFACTOR(n), unless one abandoned the bit array for a FF array and modified the sieve process to record the first factor. Something like `Bit(i) = .false.`

would be replaced by `if (FF(i) <= 1) then FF(i) = F`

Were the assignment to be made unconditionally (for faster running, perhaps), the array should be renamed to MaximumPrimeFactor. Either way, ISPRIME(n) remains easy, but much more storage than one bit per entry would be required.

If instead of the next prime one desires to find the *n* 'th prime number, then there is a difficulty that would not exist if the prime numbers only were stored in an array - but if they were then `NEXTPRIME(n)`

would have difficulty. Of course, if storage is abundant, both forms of storage could be used and each request could be handled via a simple linear index calculation into the appropriate array.

#### Supporting Prime(n)

To find the *n'*th prime number, obviously one could scan along the bit array from the start, keeping count. This will soon become tedious, so in order to support function `PRIME(n)`

, each record starts with a count of all the primes that have preceded that record's span and so to find the count for a prime fingered in that record, the scan need work only from the start of that record. So the problem reduces to determining which record is the one containing the *n* 'th prime. Since the counts are obviously strictly increasing, a binary search would be a possibility as would be an interpolating search and one could even prepare an array containing the counts so that the search could proceed without needing disc accesses - at the cost of additional storage and organisational complexity, perhaps involving Aitken's interpolation formula, except that polynomials do not provide a good fit to the required shape. Fortunately, mathematicians have considered this aspect of prime numbers also, and a rather intimidating formula is available to give an estimate of the value of the *n* 'th prime number. Equipped with this, the appropriate record can be read, the count inspected, and a scan started to find the actual *n* 'th prime. Function `PRIME(n)`

can be invoked just as array `PRIME(n)`

might be, but with something of a roil of activity in the background.

Preparing the count field involves another trick, because the first record's count field instead holds the count of records in the bit file. Now note that the sieve starts with `SORG = 3`

which means that before the first block there is a count of one prime number. Thus, if function PRIME is accessing the first block, it must know that the count of previous primes is one and not refer to the count field which instead holds the record count. When PSURGE is preparing the next batch of bits (for the second and subsequent records) it accesses the previous record to find the previous count and scans that record to count its primes so as to prepare the previous count for the new record. When the second record is being prepared by PSURGE, the previous record (the first record) has a record count of one, and, this is exactly the desired count of previous primes for the first record. But only at this point, because in moments the first record will be rewritten with a record count of two. For all this to work, `SORG = 3`

must be the case: it is not a parameter but a constant.

#### Integer Overflow

The variables are all the default thirty-two bit two's complement integers and integer overflow is a possibility, especially because someone is sure to wonder what is the largest prime that can be represented in thirty-two bits - see the output. The code would be extensible in another way, if all appearances of `INTEGER`

were to be replaced by `INTEGER*8`

though not all variables need be changed - such as `C`

and `B`

because they need only index a character in array SCHARS or a bit in a character. Using sixty-four bits for such variables is excessive even if the cpu uses a 64-bit data bus to memory. If such a change were to be made, then all would go well as cpu time and disc space were consumed up to the point when the count of prime numbers can no longer be fitted into the four character storage allowance in the record format. This will be when more than 4,294,967,295 primes have been counted (with 64-bit arithmetic its four bytes will manifest as an unsigned integer) in previous records, and Prime(4,294,967,295) = 104,484,802,043, so that the bit file would require a mere 6,530MB or so - which some may think is not too much. If so, expanding the allowance from four to five characters would be easy enough, and then 256 times as many primes could be counted. That would also expand the reach of the record counter, which otherwise would be limited to 4,294,967,295 records of 4096 bytes each, or a bit bag of 17,592,186,040,320 bytes - only seventeen terabytes...

Overflow is also a problem in many of the calculations. For instance, for a given (prime) number F, the marking of multiples of F via the sieve process starts with the bit corresponding to F² and if this exceeds the number corresponding to the last bit of the current sieve span, then the sieve process is complete for this span because all later values for F will be still further past the end. So, if `LST`

is the number corresponding to the last bit of the current span,

```
DO WHILE(F*F <= LST) !But, F*F might overflow the integer limit so instead,
DO WHILE(F <= LST/F) !Except, LST might also overflow the integer limit, so
DO WHILE(F <= (IST + 2*(SBITS - 1))/F) !Which becomes...
DO WHILE(F <= IST/F + (MOD(IST,F) + 2*(SBITS - 1))/F) !Preserving the remainder from IST/F.
```

Except, `IST`

might overflow the integer limit, in which case function PSURGE declares itself unable to proceed and returns *false*.

Overflow is detected by the sudden appearance of negative numbers, as is characteristic of two's complement integer arithmetic. This is *not* guaranteed to be used on all computers (notably, on a decimal computer such as the IBM1620 and others), and in its absence, the procedure will malfunction. Some systems detect integer overflow via hardware (a special "flag" register, or an interrupt) and there may be facilities for noticing such events. First Fortran (1957) offered special statements such as `IF ACCUMULATOR OVERFLOW `

(yes, without brackets) and similarly for QUOTIENT OVERFLOW and DIVIDE CHECK but they were abandoned by the modernisers. The only general solution to this problem would be to convert to using multiple-precision (or "bignum") arithmetic, whereupon the code becomes extensible in another way simply by extending as needed the amount of storage allowed for variables.
*labelon*,*labeloff*

These methods have been tested by converting `INTEGER`

to `INTEGER*2`

and also by using a record size of sixteen bytes (because UltraEdit, when displaying in binary, shows that many bytes to a line), and it was a real pleasure for once to be able to read the 32-bit count field at the start of each record left-to-right in hexadecimal rather than in the crazed little-endian order that would otherwise have been used.

### The Code

The source code employs the MODULE facility of F90 simply to avoid the tedium of setting up a COMMON storage area and having to declare the type of the functions in every routine that uses them. Otherwise, older style compilers will accept this, except for an occasional array facility (such as `BIT8 = CHAR(255)`

) and the use of the $ format code to allow the next output to tag on to the same line. The rather more fearsome declaration `RECURSIVE FUNCTION NEXTPRIME`

could be avoided if in subroutine PSURGE, the invocation of NEXTPRIME was replaced by in-line code. Similarly with subroutine GETSREC, though forgetting this didn't seem to make any difference. This is just convenience recursion, not structural recursion to some arbitrary depth.

Although recursion is now permissible if one utters the magic word `RECURSIVE`

, this ability usually is not extended to the workings for formatted I/O so that if say a function is invoked in a WRITE statement's list, should that function attempt to use a WRITE statement, the run will be stopped. There can be slight dispensations if different types of WRITE statement are involved (say, formatted for one, and "free"-format for the other) but an ugly message is the likely result. The various functions are thus best invoked via an assignment statement to a scratch variable, which can then be printed. The functions are definitely not "pure" because although they are indeed functions only of their arguments, they all mess with shared storage, can produce error messages (and even a STOP), and can provoke I/O with a disc file, even creating such a file. For this reason, it would be unwise to attempt to invoke them via any sort of parallel processing. Similarly, the disc file is opened with exclusive use because of the possibility of writing to it. There are no facilities in standard Fortran to control the locking and unlocking of records of a disc file as would be needed when adding a new record and updating the record count. This would be needed if separate tasks could be accessing the bit file at the same time, and is prevented by exclusive use. If an interactive system were prepared to respond to requests for ISPRIME(n), *etc.* it should open the bit file only for its query then close it before waiting for the next request - which might be many milliseconds away.

The bit array is stored in an array of type CHARACTER*1 since this has been available longer and more widely than INTEGER*1. One hopes that the consequent genuflections to type checking via functions CHAR(i) and ICHAR(c) will not involve an overhead.

```
MODULE PRIMEBAG !Need prime numbers? Plenty are available.
C Creates and expands a disc file for a sieve of Eratoshenes, representing odd numbers only and starting with three.
C Storage requirements: an array of N prime numbers in 16/32/64 bits vs. a bit array up to the 16/32/64 bit limit.
C Word size N Prime N words in bits Bit array in bits.
C 8 bit P(31) = 127 248 128
C P(54) = 251 432 256
C 16 bit P(3,512) = 32,749 56,192 32,768
C P(6,542) = 65,521 104,672 65,536
C 32 bit P(105,097,565) = 2,147,483,647 3,363,122,080 2,147,483,648
C P(203,280,221) = 4,294,967,291 6,504,967,072 4,294,967,296
C 64 bit 2.112E17 ? 1.352E19 9,223,372,036,854,775,808 ~ 9.22E18
C from n/Ln(n) 4.158E17 ? 2.661E19 18,446,744,073,709,551,616 ~ 1.84E19
INTEGER MSG !I/O unit number.
INTEGER SSTASH !For attachment to my stash file.
INTEGER SRECLEN,SCHARS,SBITS !Sizes.
INTEGER SORG !Where the sieve starts. This must be three.
INTEGER SLAST !Last record in my stash file.
DATA SSTASH,SREC,SLAST/0,0,0/ !Prepared by PRIMEBAG.
PARAMETER (SRECLEN = 1024) !4K disc bloc size, but RECL (in OPEN) is in terms of four-byte integers.
PARAMETER (SCHARS = (SRECLEN - 1)*4) !Reserving space for one number at the start.
PARAMETER (SBITS = SCHARS*8) !Known size of a character.
PARAMETER (SORG = 3) !First odd number past two, which is not odd.
CHARACTER*(*) SFILE !A name is needed.
PARAMETER (SFILE = "C:/Nicky/RosettaCode/Primes/PrimeSieve.bit") !I don't have to count the characters.
Components of a buffered record for the stash.
INTEGER SREC !The record number.
CHARACTER*1 C4(4) !The start of the record - a counter.
CHARACTER*1 SCHAR(0:SCHARS - 1) !The majority of the record - a bit array, packed in 8-bit blobs...
Collect some bit twiddling assistants for AND and OR, rather than bit shifting.
CHARACTER*1 BITON(0:7),BITOFF(0:7) !Functions IBSET and IBCLR may not be available, and are little-endian anyway.
PARAMETER (BITON =(/CHAR(2#10000000),CHAR(2#01000000), !128, 64, Reading strictly left-to-right.
1 CHAR(2#00100000),CHAR(2#00010000), ! 32, 16, Uncompromising bigendery.
1 CHAR(2#00001000),CHAR(2#00000100), ! 8, 4, Not just for bytes in words,
3 CHAR(2#00000010),CHAR(2#00000001)/)) ! 2, 1. But also bits in bytes.
PARAMETER (BITOFF=(/CHAR(2#01111111),CHAR(2#10111111), !127, 191, BITON + BITOFF = 255.
2 CHAR(2#11011111),CHAR(2#11101111), !223, 239,
1 CHAR(2#11110111),CHAR(2#11111011), !247, 251,
3 CHAR(2#11111101),CHAR(2#11111110)/)) !253, 254.
CONTAINS
INTEGER FUNCTION I4UNPACK(C4) !Convert four successive characters into an integer.
CHARACTER*1 C4(4) !The characters.
I4UNPACK = ((ICHAR(C4(1))*256 + ICHAR(C4(2)))*256 !Convert the first four bytes
1 + ICHAR(C4(3)))*256 + ICHAR(C4(4)) !To a four-byte integer.
END FUNCTION I4UNPACK !Big-endian style, irrespective of cpu endianness.
SUBROUTINE C4PACK(I4) !Convert an integer into successive bytes.
Could return the result via a fancy function, but for now a global variable will do.
INTEGER I4,N !The integer, and a copy to damage.
INTEGER I !A stepper.
N = I4 !Keep the original safe.
DO I = 4,1,-1 !Know that four characters will do. Fixed format makes this easy.
C4(I) = CHAR(MOD(N,256)) !Grab the low-order eight bits.
N = N/256 !And shift right eight.
END DO !Do it again.
END SUBROUTINE C4PACK !Stored big-endianly, irrespective of cpu endianness.
LOGICAL FUNCTION GRASPPRIMEBAG(F)
INTEGER F !The I/O unit number to use.
LOGICAL EXIST !Use the keyword as a name
INTEGER IOSTAT !And don't worry over assignment direction.
CHARACTER*3 STYLE !One way or another.
SSTASH = F !I shall use it.
INQUIRE (FILE = SFILE,EXIST = EXIST) !Trouble with a missing "path" may arise.
IF (EXIST) THEN !If the file exists,
STYLE = "OLD" !I shall read it.
ELSE !But if it doesn't,
STYLE = "NEW" !I shall create it.
END IF !Enough prevarication.
OPEN(SSTASH,FILE = SFILE, STATUS = STYLE, !Go for the file.
& ACCESS = "DIRECT", RECL = SRECLEN, FORM = "UNFORMATTED", !I have plans.
& ERR = 666, IOSTAT = IOSTAT) !Which may be thwarted.
IF (EXIST) THEN !If there is one...
CALL READSCHAR(1) !The first record is also a header.
SLAST = I4UNPACK(C4) !The number of records stored.
ELSE !Otherwise, start from scratch.
SLAST = 0 !No saved records.
CALL PSURGE(SCHAR) !During preparation of the first batch of bits.
END IF !All should now be in readiness.
GRASPPRIMEBAG = .TRUE.!So, feel confidence.
RETURN !And escape.
666 WRITE (*,667) IOSTAT,SFILE !But, something may have gone wrong.
667 FORMAT ("Pox! Error code ",I0, !A "hole" in the directory path?
1 " when attempting to open file ",A) !Read-only access allowed when I want "update"?
GRASPPRIMEBAG = .FALSE. !Whatever, it didn't work.
END FUNCTION GRASPPRIMEBAG !So much for that.
SUBROUTINE READSCHAR(R) !Get record R into SCHAR, which may already hold it.
INTEGER R !The record number desired.
IF (R.EQ.SREC) RETURN !Perhaps it is already to hand.
SREC = R !If not, move attention to it.
READ (SSTASH,REC = SREC) C4,SCHAR !And read the record.
END SUBROUTINE READSCHAR!Thus, I have a buffer too.
LOGICAL FUNCTION PSURGE(BIT8) !Add another record to the stash.
C Surges forward into the next batch of primes, to be stored via a bit array in the file.
C Each record starts with a count of the number of primes that have gone before.
C Except that for the first record, this is the record counter for the stash file.
C Except that when starting the second record, one is also the number of primes before SORG.
CHARACTER*1 BIT8(0:SCHARS - 1) !Watch out! This may be SCHAR itself!
INTEGER IST,LST !The numbers spanned by the surge.
INTEGER F !A factor.
INTEGER I !Another factor and a stepper.
INTEGER C !Index for array BIT8.
INTEGER NP !Number of primes.
Carry forward the count of previous primes to start the following record..
10 IF (SLAST.GT.0) THEN !Is there a previous record?
CALL READSCHAR(SLAST) !Yes. Grab it. A good chance this is already in C4,SCHAR.
NP = I4UNPACK(C4) !Its count of the primes accumulated before it.
DO I = 0,SCHARS - 1 !Find out how namy primes it fingered by scanning its bits.
NP = NP + COUNT(IAND(ICHAR(SCHAR(I)),ICHAR(BITON)).NE.0) !Whee! Eight at a go!
END DO !On to the next byte.
END IF !When creating a new record, its follower may not be sought in this run.
Concoct the next batch of bits. Contorted calculations avoid integer overflow.
20 BIT8 = CHAR(255) !All bits are aligned with numbers that might prove to be prime.
IST = SORG + SLAST*(2*SBITS) !Bit(0) of BIT8(0) corresponds to IST.
LST = IST + 2*(SBITS - 1) !Bit(last) to this number. Remember, only odd numbers have bits.
IF (IST.LE.0) THEN !Humm. I'd better check.
WRITE (MSG,21) SLAST,IST,LST !This works only with two's complement integers.
21 FORMAT (/,"Integer overflow in the sieve of Eratosthenes!", !Oh dear.
1 /,"Advancing from surge ",I0," to span ",I0," to ",I0) !These numbers will look odd.
PSURGE = .FALSE. !But it is better than no indication of what went wrong.
RETURN !Give in.
END IF !Enough worrying.
F = 3 !The first possible factor. Zapping will start at F²
c DO WHILE(F.LE.LST/F) !If F² is past the end, so will be still larger F: enough.
DO WHILE(F.LE.IST/F + (MOD(IST,F) + 2*(SBITS - 1))/F) !"Synthetic division" avoiding overflow.
I = (IST - 1)/F + 1 !I want the first multiple of F in IST:LST. F may be a factor of IST.
IF (MOD(I,2).EQ.0) I = I + 1!If even, advance to the next odd multiple. Even numbers are omitted by design.
IF (I.LT.F) I = F !Less than F is superfluous: the position was zapped by earlier action.
c I = (I*F - IST)/2 !Current bit positions are for IST, IST+2, IST+4, etc.
I = ((I - IST/F)*F - MOD(IST,F))/2 !Avoids overflow when calculating the start value, I*F.
DO I = I,SBITS - 1,F !Zap every F'th bit along. This is the sieve of Eratosthenes.
C = I/8 !Eight bits per character.
BIT8(C) = CHAR(IAND(ICHAR(BIT8(C)), !For F = 3 and 5, characters will be hit more than once.
1 ICHAR(BITOFF(MOD(I,8))))) !Whack a bit. All the above just for this!
END DO !On to the next bit.
22 F = NEXTPRIME(F) !So much for F. Next, please.
END DO !Are we there yet?
Correct the count in the header, if this is an added record.
30 IF (SLAST.GT.0) THEN !So, was there a pre-existing header record?
CALL READSCHAR(1) !Yes. Get the header record into C4,SCHAR.
CALL C4PACK(SLAST + 1) !This is the new record count.
WRITE (SSTASH,REC = 1) C4,SCHAR !Write it all back.
SCHAR = BIT8 !Ensure that SCHAR and SREC will be agreed.
END IF !So much for the header's count.
Cast the bits into the stash by writing record SLAST + 1..
40 IF (SLAST.EQ.0) THEN !If we're writing the first record,
CALL C4PACK(1) !Then this is the record count.
ELSE !Otherwise,
CALL C4PACK(NP) !Place the previous primes count.
END IF !All this to help PRIME(i).
SLAST = SLAST + 1 !This is now the last stashed record.
WRITE (SSTASH,REC = SLAST) C4,BIT8 !I/O directly from the work area?
SREC = SLAST !This is where BIT8 was written.
PSURGE = .TRUE. !That assumes BIT8 is not SCHAR for SLAST > 1.
END FUNCTION PSURGE !That was fun!
RECURSIVE SUBROUTINE GETSREC(R) !Make present the bit array belonging to record R.
INTEGER R !The record number..
CHARACTER*1 BIT8(0:SCHARS - 1) !A scratchpad. Others may be relying on SCHAR.
IF (SLAST.LE.0) RETURN!DANGER! The first record is being initialised!
DO WHILE(SLAST.LT.R) !If we haven't reached so far,
IF (.NOT.PSURGE(BIT8)) THEN !Slog forwards one record's worth.
WRITE (MSG,1) R !Or maybe not.
1 FORMAT ("Cannot prepare surge ",I0) !Explain.
STOP "No bits, no go." !And quit.
END IF !And having prepared the next block of bits,
END DO !Check afresh.
CALL READSCHAR(R) !Read the desired record's bits.
END SUBROUTINE GETSREC !Done.
INTEGER FUNCTION PRIME(N) !P(1) = 2, P(2) = 3, etc.
C Calculate P(n) ~ n.ln(n)
C ~ n{ln(n) + ln(ln(n)) - 1 + (ln(ln(n)) - 2)/ln(n) - [ln(ln(n))**2 - 6*log(log(n)) + 11]/[2*(ln(n))**2] + ....}
C J.B.Rosser's 1938 Theorem: n[ln(n) + ln(ln(n)) - 1] < P(n) < n[ln(n) + ln(ln(n))]
C or, with E = ln(n) + ln(ln(n)), n[E - 1] < P(n) < n[E]
C Experimentation shows that the undershoot of the first two terms involves many records worth of bits.
C Including additional terms does much better, but can overshoot.
INTEGER N !The desired one.
INTEGER R,NP !Counts.
INTEGER B,C !Bit and character indices.
DOUBLE PRECISION EST,LN,LLN !Hope, if not actuality.
IF (N.LE.0) STOP "Primes are counted positively!" !Something must be wrong!
IF (N.LE.1) THEN !The start of the bit array being preempted.
PRIME = 2 !So, no array access.
ELSE !Otherwise, the fun begins.
LN = LOG(DFLOAT(N)) !Here we go.
LLN = LOG(LN) !A popular term.
EST = N*(LN !Estimate the value of the N'th prime.
1 + LLN - 1 !Second term
2 + (LLN - 2)/LN !Third term.
3 - (LLN**2 - 6*LLN + 11)/(2*LN**2)) !Fourth term.
R = (EST - SORG)/(2*SBITS) + 1 !Thereby selecting a record to scan.
IF (R.LE.0) R = 1 !And not making a mess with N < 6 or so.
9 CALL GETSREC(R) !Go for the record.
IF (R.LE.1) THEN !The first record starts with the record count.
NP = 1 !And I know how many primes precede its start point
ELSE !While for all subsequent records,
NP = I4UNPACK(C4) !This counts the number of primes that precede record R's start number.
END IF !So now I'm ready to count onwards.
IF (N.LE.NP) THEN !Maybe not.
R = R - 1 !The estimate took me too far ahead.
GO TO 9 !Try again.
END IF !Could escalate to a binary search or even an interpolating search.
Commence scanning the bits.
C = 0 !Start with the first character of SREC..
B = -1 !Syncopation. The formula is known to always under-estimate.
10 IF (NP.LT.N) THEN !Are we there yet?
11 B = B + 1 !No. Advance to the next bit.
IF (B.GE.8) THEN !Overflowed a character yet?
B = 0 !Yes. Start afresh at the first bit.
C = C + 1 !And advance one character.
IF (C.GE.SCHARS) THEN !Overflowed the record yet?
C = 0 !Yes. Start afresh at its first character.
R = R + 1 !And advance to the next record.
CALL GETSREC(R) !Possibly, create it.
END IF !So much for records.
END IF !We're now ready to test bit B of character C of record R.
IF (IAND(ICHAR(SCHAR(C)),ICHAR(BITON(B))).EQ.0) GO TO 11 !Not a prime. Search on.
NP = NP + 1 !Count another prime.
GO TO 10 !Pehaps this will be the one.
END IF !So much for the search.
PRIME = SORG + (R - 1)*(2*SBITS) + (C*8 + B)*2 !The corresponding number.
IF (PRIME.LE.0) WRITE (MSG,666) N,PRIME !Or, possibly not.
666 FORMAT ("Integer overflow! Prime(",I0,") gives ",I0,"!") !Let us hope the caller notices.
END IF !So, all going well,
END FUNCTION PRIME !It is found.
RECURSIVE INTEGER FUNCTION NEXTPRIME(N) !Keep right on to the end of the road.
Can invoke GETSREC, which can invoke PSURGE, which ... invokes NEXTPRIME. Oh dear.
INTEGER N !Not necessarily itself a prime number.
INTEGER NN !A value to work with.
INTEGER R !A record number into the stash.
INTEGER I,IST !Number offsets.
INTEGER C,B !Character and bit index.
IF (N.LE.1) THEN !Suspicion prevails.
NN = 2 !This is not represented in my bit array.
ELSE !Otherwise, the fun begins.
NN = N + 1 !Advance, with a copy I can mess with.
IF (MOD(NN,2).EQ.0) NN = NN + 1 !Thus, NN is now odd.
IF (NN.LE.0) GO TO 666 !But perhaps not proper, due to overflow.
R = (NN - SORG)/(2*SBITS) !SORG is odd, so (NN - SORG) is even.
CALL GETSREC(R + 1) !The first record is numbered one, not zero.
IST = SORG + R*(2*SBITS) !The number for its first bit: even numbers are omitted..
I = (NN - IST)/2 !Offset into the record. NN - IST is even.
C = I/8 !Which character in SCHAR(0:SCHARS - 1)?
B = MOD(I,8) !Which bit in SCHAR(C)?
10 IF (IAND(ICHAR(SCHAR(C)),ICHAR(BITON(B))).EQ.0) THEN !On for a prime.
NN = NN + 2 !Alas, it is off, so NN is not a prime. Perhaps this will be.
B = B + 1 !Advance one bit. Each bit steps two.
IF (B.GE.8) THEN !Past the end of the character?
B = 0 !Yes. Back to bit zero.
C = C + 1 !And advance one chracter.
IF (C.GE.SCHARS) THEN !Past the end of the record?
IF (NN.LE.0) GO TO 666!Yes. If NN has overflowed, the end of the rope is reached.
C = 0 !Back to the start of a record.
R = R + 1 !Advance one record.
CALL GETSREC(R + 1) !And read it. (Count is from 1, not 0).
END IF !So much for overflowing a record.
END IF !So much for overflowing a character.
GO TO 10 !Try again.
END IF !So much for the bit array.
END IF !If there had been a scan.
NEXTPRIME = NN !The number for which the scan stopped.
IF (NN.GT.0) RETURN !All is well.
666 WRITE (MSG,667) N,NN !Or, maybe not. Careful: this won't appear if NEXTPRIME is invoked in a WRITE list.
667 FORMAT ("Integer overflow! NextPrime(",I0,") gives ",I0,"!") !The recipient could do a two's complement.
NEXTPRIME = NN !Prefer to return the bad value rather than fail to return anything.
END FUNCTION NEXTPRIME !No divisions, no sieving. Here, anyway
INTEGER FUNCTION PREVIOUSPRIME(N) !If N is good, this can't overflow.
INTEGER N !The number, not necessarily a prime.
INTEGER NN !A value to mess with.
INTEGER R !A record number.
INTEGER I !Offset.
INTEGER C,B !Character and bit fingers.
IF (N.LE.3) THEN !Suppress annoyances.
NN = 2 !This is now called the first prime, not one.
ELSE !Otherwise, some work is to be done.
NN = N - 1 !Step back one to ensure previousness.
IF (MOD(NN,2).EQ.0) NN = NN - 1 !And here, oddness is a minimal requirement.
R = (NN - SORG)/(2*SBITS) !Finger the record containing the bit for NN.
CALL GETSREC(R + 1) !Record counting starts with one.
I = (NN - (SORG + R*(2*SBITS)))/2 !Offset into that record.
C = I/8 !Finger the character in SCHAR.
B = MOD(I,8) !And the bit within the character.
10 IF (IAND(ICHAR(SCHAR(C)),ICHAR(BITON(B))).EQ.0) THEN !On for a prime.
NN = NN - 2 !Alas, it is off, so NN is not a prime. Perhaps this will be.
B = B - 1 !Retreat one bit. Each bit steps two.
IF (B.LT.0) THEN !Past the start of the character?
B = 7 !Yes. Back to the last bit.
C = C - 1 !And retreat one chracter.
IF (C.LT.0) THEN !Past the start of the record?
C = SCHARS - 1 !Yes. Back to the end of a record.
R = R - 1 !Retreat one record.
CALL GETSREC(R + 1) !And read it. (Count is from 1, not 0).
END IF !So much for overflowing a record.
END IF !So much for overflowing a character.
GO TO 10 !Try again.
END IF !So much for the bit array.
END IF !Possibly, it was not needed.
PREVIOUSPRIME = NN !There.
END FUNCTION PREVIOUSPRIME !Doesn't overflow, either.
LOGICAL FUNCTION ISPRIME(N) !Could fool around explicity testing 2 and 3 and say 5,
INTEGER N !But that means also checking that N > 2, N > 3, and N > 5.
c ISPRIME = N .EQ. NEXTPRIME(N - 1) !This is so much easier, but involves scanning to reach the next prime.
INTEGER R,IST,I,C,B !Assistants for indexing the bit array.
IF (N.LE.1) THEN !First, preclude sillyness.
ISPRIME = .FALSE. !Not a prime.
ELSE IF (N.EQ.2) THEN !This is the only even number
ISPRIME = .TRUE. !That is a prime.
ELSE IF (MOD(N,2).EQ.0) THEN !Other even numbers
ISPRIME = .FALSE. !Are not prime numbers.
ELSE !Righto, now N is an odd number and there is a bit array for them.
R = (N - SORG)/(2*SBITS) !SORG is odd, so (N - SORG) is even.
CALL GETSREC(R + 1) !The first record is numbered one, not zero.
IST = SORG + R*(2*SBITS) !The number for its first bit: even numbers are omitted.
I = (N - IST)/2 !Offset into the record. N - IST is even.
C = I/8 !Which character in SCHAR(0:SCHARS - 1)?
B = MOD(I,8) !Which bit in SCHAR(C), indexing from zero?
ISPRIME = IAND(ICHAR(SCHAR(C)),ICHAR(BITON(B))).GT.0 !The bit is on for a prime.
END IF !All that fuss to find a single bit.
END FUNCTION ISPRIME !But, no divisions up to SQRT(N) or the like.
END MODULE PRIMEBAG !Functions updating a disc file as a side effect...
PROGRAM POKE
USE PRIMEBAG
INTEGER I,P,N,N1,N2 !Assorted assistants.
INTEGER ORDER !A collection of special values.
PARAMETER (ORDER = 6) !For one, two, and four byte integers.
INTEGER EDGE(ORDER) !Considered as two's complement and unsigned.
PARAMETER (EDGE = (/31,54,3512,6542,105097565,203280221/)) !These primes are of interest.
MSG = 6 !Standard output.
IF (.NOT.GRASPPRIMEBAG(66)) STOP "Gan't grab my file!" !Attempt in hope.
Case 1.
C FORALL(I = 1:20) LIST(I) = PRIME(I) is rejected because function Prime(i) is rather impure.
10 WRITE (MSG,11)
11 FORMAT (19X,"First twenty primes: ", $)
DO I = 1,20
P = PRIME(I)
WRITE (MSG,12) P
12 FORMAT (I0,",",$)
END DO
Case 2.
20 WRITE (MSG,21)
21 FORMAT (/,12X,"Primes between 100 and 150: ",$)
P = 100
22 P = NEXTPRIME(P) !While (P:=NextPrime(P)) <= 150 do Print P;
IF (P.LE.150) THEN !But alas, no assignment within an expression.
WRITE (MSG,23) P
23 FORMAT (I0,",",$)
GO TO 22
END IF
Case 3.
30 N1 = 7700 !Might as well parameterise this.
N2 = 8000 !Rather than litter the source with explicit integers.
N = 0
P = N1
31 P = NEXTPRIME(P)
IF (P.LE.N2) THEN
N = N + 1
GO TO 31
END IF
WRITE (MSG,32) N1,N2,N
32 FORMAT (/"Number of primes between ",I0," and ",I0,": ",I0)
Case 4.
40 WRITE (MSG,41)
41 FORMAT (/,"Tenfold steps...")
N = 1
DO I = 1,9 !This goes about as far as it can go.
P = PRIME(N)
WRITE (MSG,42) N,P
42 FORMAT ("Prime(",I0,") = ",I0)
N = N*10
END DO
Cast forth some interesting values.
100 WRITE (MSG,101)
101 FORMAT (/,"Primes close to number sizes")
DO N = 1,ORDER !Step through the list.
N1 = EDGE(N) - 1 !Syncopation for the special value.
DO I = 1,2 !I want the prime on either side.
N1 = N1 + 1 !So, there are two successive primes to finger.
WRITE (MSG,102) N1 !Identify the index.
102 FORMAT ("Prime(",I0,") = ",$) !Piecemeal writing to the output,
P = PRIME(N1) !As this may fling forth a complaint.
WRITE (MSG,103) P !Show the value returned.
103 FORMAT (I0,", ",$) !Which may be unexpected.
END DO !On to the second.
WRITE (MSG,*) !End the line after the second result.
END DO !On to the next in the list.
END !Whee!
```

Although the structuralist talk up the merit of "structured" constructs in programming, there are often annoying obstacles. With a WHILE-loop one usually has to repeat the "next item" code to "prime" the loop, as in

```
P = NEXTPRIME(100)
DO WHILE (P.LE.150)
...stuff...
P = NEXTPRIME(P)
END DO
```

While this is not a large imposition in this example, if Fortran were to allow assignment within expressions as in Algol, the tedium of code replication and its risks could be avoided.

` P:=100; WHILE (P:=NextPrime(P)) <= 150 DO stuff;`

If instead of NEXTPRIME the "next item" code was to read a record of a disc file, the repetition needed becomes tiresome. So, an IF-statement, and ... a GO TO...

### The Results

When creating the bit file, everything appears in a blink up to the millionth prime, then a pause until the ten millionth prime, then about a minute to attain the hundred millionth prime. The blinking lights show that there is not much disc I/O in progress (actually, a solid-state unit) while the cpu is running at full speed. There is a further pause from there until overflow is reached. For a subsequent run with the disc file already prepared, all is completed in a blink. Pausing the Great Internet Mersenne Prime crunch (sixfold) increases the speed and I/O rate by about a third. During the file expansion, the rate of I/O slowly decreases at a decreasing rate - this is due to each successive sieve surge requiring more primes to step with, but the primes themselves thin out and being larger, require fewer steps to traverse the span. The effort for each surge is related to the sum of the reciprocals of the primes that will sieved with and this forms an interesting sequence in its own right. For a given surge width, fewer and fewer hits are made within that width by the larger prime numbers. However, 4092 bytes (four are reserved for the count, and on this windows XP system disc space is allocated in blocks of 4k) gives 32736 bits which with odd numbers only, spans 65472. With 32-bit two's complement, sqrt(2,147,483,647) = 46340·95 so even the largest stepper will land at least once within every span, except that only odd multiples are involved so already the hit rate has drifted below one per span and with extension to still larger numbers, the hit rate will fall further. Thus, if instead each pass were to be for the full width of the bit array in the disc file, the I/O system would suffer a thrashing during the multiple passes unless the entire file could be fitted into random-access memory. But such a scheme would not have the "extensible" aspect.

Output:

First twenty primes: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71, Primes between 100 and 150: 101,103,107,109,113,127,131,137,139,149, Number of primes between 7700 and 8000: 30 Tenfold steps... Prime(1) = 2 Prime(10) = 29 Prime(100) = 541 Prime(1000) = 7919 Prime(10000) = 104729 Prime(100000) = 1299709 Prime(1000000) = 15485863 Prime(10000000) = 179424673 Prime(100000000) = 2038074743 Primes close to number sizes Prime(31) = 127, Prime(32) = 131, Prime(54) = 251, Prime(55) = 257, Prime(3512) = 32749, Prime(3513) = 32771, Prime(6542) = 65521, Prime(6543) = 65537, Prime(105097565) = 2147483647, Prime(105097566) = Integer overflow! Prime(105097566) gives -2147483637! -2147483637, Prime(203280221) = Integer overflow in the sieve of Eratosthenes! Advancing from surge 32801 to span -2147420221 to -2147354751 Cannot prepare surge 65601 No bits, no go.

The disc file holding all primes up to the thirty-two bit limit occupies 134,352,896 bytes, or 128MB. Nothing much, these days. Activating 7-zip out of curiosity resulted in compressing the file by a factor of two. As the primes thin out there will be more and more characters with only one bit on (if not none) rather than a fuller selection. However, the existence of prime pairs shows that the bits will never be all lonely forever. For this run, the last record is number 32,801 and 105,097,477 (hex 643A905) primes have gone before.

First value Bit array... 3: 11101101 10100110 01011010 01001100 10110010 10010001 01101101 00000010 10011000 01100100 10100100 11000011 01100000 10000010 11010011 00001001 00100110 01011000 01000000 10110100 00001001 00001101 00100010 01001010 01000101 00010000 11000011 00101001 00010110 10000010 00101000 10100100 ... 2147481603: 00000000 00000100 00000000 00000000 00010000 00000000 00000000 00000000 00000000 00000000 00000000 00000001 01000000 10000000 10000000 00000000 00100000 00001000 01001100 10000000 00000001 00000100 00000010 00000000 00000100 00000000 01000000 00000000 00000010 00000001 00001100 00000000 ...

Would anyone prefer to see that bit array in the little-endian order within bytes?

The thinning out rather suggests an alternative encoding such as by counting the number of "off" bits until the next "on" bit, but this would produce a variable-length packing so that it would no longer be easy to find the bit associated with a given number by something so simple as `R = (NN - SORG)/(2*SBITS)`

as in NEXTPRIME. A more accomplished data compression system might instead offer a reference to a library containing a table of prime numbers, or even store the code for a programme that will generate the data. File Pbag.for is 23,008 bytes long, and of course contains irrelevant commentary and flabby phrases such as "PARAMETER", "INTEGER", "GRASPPRIMEBAG", *etc*. As is the modern style, the code file is much larger at 548,923 bytes (containing kilobyte sequences of hex CC and of 00), but both are much smaller than the 134,352,896 bytes of file PrimeSieve.bit.

### Alternative Version using the Sorenson Sieve

This version is written in largely Fortran-77 style (fixed form, GO TO) It uses the dynamic array algorithm as described in the paper, "Two Compact Incremental Prime Sieves" by Jonathon P. Sorenson. The sieve is limited to an upper bound of 2^31 (in Fortran, integers are always signed) To be correct for that upper limit, signed overflow that can occur for items in the buckets (8 16-bit signed integers) must be taken into account.

This code is also written in old style in that the generator uses statically allocated variables to maintain state. It is not re-entrant and there can only be one generator in use at a time. That's more in line with programming practices in the 70s and 80s.

In Fortran, dynamic arrays are always exactly allocated without any over capacity for later expansion, so naively re-allocating the array by 2 as specified in the original paper results in a significant drop in performance due to all of the memory moves. Therefore, this implementation keeps track of the array size which will be less than the array capacity.

```
* incremental Sieve of Eratosthenes based on the paper,
* "Two Compact Incremental Prime Sieves"
SUBROUTINE nextprime(no init, p)
IMPLICIT NONE
INTEGER*2, SAVE, ALLOCATABLE :: sieve(:,:)
INTEGER, SAVE :: r, s, pos, n, f1, f2, sz
INTEGER i, j, d, next, p, f3
LOGICAL no init, is prime
IF (no init) GO TO 10
IF (ALLOCATED(sieve)) DEALLOCATE(sieve)
* Each row in the sieve is a stack of 8 short integers. The
* stacks will never overflow since the product 2*3*5 ... *29
* (10 primes) exceeds a 32 bit integer. 2 is not stored in the sieve.
ALLOCATE(sieve(8,3))
sieve = reshape([(0_2, i = 1, 24)], shape(sieve))
r = 3
s = 9
pos = 1
sz = 1 ! sieve starts with size = 1
f1 = 2 ! Fibonacci sequence for allocating new capacities
f2 = 3 ! array starts with capacity 3
n = 1
p = 2 ! return our first prime
RETURN
10 n = n + 2
is prime = .true.
IF (sieve(1, pos) .eq. 0) GO TO 20 ! n is non-smooth w.r.t sieve
is prime = .false. ! element at sieve(pos) divides n
DO 17, i = 1, 8
Clear the stack of divisors by moving them to the next multiple
d = sieve(i, pos)
IF (d .eq. 0) GO TO 20 ! stack is empty
IF (d .lt. 0) d = d + 65536 ! correct storage overflow
sieve(i, pos) = 0
next = mod(pos + d - 1, sz) + 1
* Push divisor d on to the stack of the next multiple
j = 1
12 IF (sieve(j, next) .eq. 0) GO TO 15
j = j + 1
GO TO 12
15 sieve(j, next) = d
17 CONTINUE
Check if n is square; if so, then add sieving prime and advance
20 IF (n .lt. s) GO TO 30
IF (.not. is prime) GO TO 25
is prime = .false. ! r = √s divides n
next = mod(pos + r - 1, sz) + 1 ! however, r is prime, insert it.
j = 1
22 IF (sieve(j, next) .eq. 0) GO TO 23
j = j + 1
GO TO 22
23 sieve(j, next) = r
25 r = r + 2
s = r**2
Continue to the next array slot; grow the array by two when
* we get to the end to maintain the invariant size(sieve) > √n
* IF the size exceeds the array capacity, resize the arary.
30 pos = pos + 1
IF (pos .le. sz) GO TO 40
sz = sz + 2
pos = 1
IF (sz .le. f2) GO TO 40 ! so far, no need to grow
f3 = f1 + f2
f1 = f2
f2 = f3
sieve = reshape(sieve, [8, f2],
& pad = [(0_2, i = 1, 8*(f2 - f1))])
* Either return n back to the caller or circle back if n
* turned out to be composite.
40 IF (.not. is prime) GO TO 10
p = n
END SUBROUTINE
```

The driver code:

```
INCLUDE 'sieve.f'
PROGRAM RC Extensible Sieve
IMPLICIT INTEGER (A-Z)
WRITE (*, '(A)', advance='no')
& 'The first 20 primes:'
CALL nextprime(.false., p)
DO 10, i = 1, 20
WRITE (*, '(I3)', advance = 'no') p
10 CALL nextprime(.true., p)
WRITE (*, *)
WRITE (*, '(A)', advance = 'no')
& 'The primes between 100 and 150:'
20 CALL nextprime(.true., p)
IF (p .gt. 149) GO TO 30
IF (p .gt. 99)
& WRITE (*, '(I4)', advance = 'no') p
GO TO 20
30 WRITE (*, *)
count = 0
40 CALL nextprime(.true., p)
IF (p .gt. 7999) GO TO 50
IF (p .gt. 7700) count = count + 1
GO TO 40
50 WRITE (*, 100) count
100 FORMAT ('There are ', I0, ' primes between 7700 and 8000.')
CALL nextprime(.false., p) ! re-initialize
target = 1 ! target count
n = 0 ! number of primes generated
60 n = n + 1
IF (n .lt. target) GO TO 70
WRITE (*, '(ES7.1,1X,I12)'), real(n), p
IF (target .eq. 100 000 000) GO TO 80
target = target * 10
70 CALL nextprime(.true., p)
GO TO 60
80 END
```

- Output:

The first 20 primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 The primes between 100 and 150: 101 103 107 109 113 127 131 137 139 149 There are 30 primes between 7700 and 8000. 1.0E+00 2 1.0E+01 29 1.0E+02 541 1.0E+03 7919 1.0E+04 104729 1.0E+05 1299709 1.0E+06 15485863 1.0E+07 179424673 1.0E+08 2038074743

## Frink

Frink has built-in functions for efficiently enumerating through prime numbers, including `primes`

, `nextPrime`

, and `previousPrime`

, and `isPrime`

. These functions handle arbitrarily-large integers.

```
println["The first 20 primes are: " + first[primes[], 20]]
println["The primes between 100 and 150 are: " + primes[100,150]]
println["The number of primes between 7700 and 8000 are: " + length[primes[7700,8000]]]
println["The 10,000th prime is: " + nth[primes[], 10000-1]] // nth is zero-based
```

- Output:

The first 20 primes are: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71] The primes between 100 and 150 are: [101, 103, 107, 109, 113, 127, 131, 137, 139, 149] The number of primes between 7700 and 8000 are: 30 The 10,000th prime is: 104729

## FutureBasic

```
local fn IsPrime( n as NSUInteger ) as BOOL
BOOL isPrime = YES
NSUInteger i
if n < 2 then exit fn = NO
if n = 2 then exit fn = YES
if n mod 2 == 0 then exit fn = NO
for i = 3 to int(n^.5) step 2
if n mod i == 0 then exit fn = NO
next
end fn = isPrime
local fn ExtensiblePrimes
long c = 0, n = 2, count = 0, track = 0
printf @"The first 20 prime numbers are: "
while ( c < 20 )
if ( fn IsPrime(n) )
printf @"%ld \b", n
c++
end if
n++
wend
printf @"\n\nPrimes between 100 and 150 include: "
for n = 100 to 150
if ( fn IsPrime(n) ) then printf @"%ld \b", n
next
printf @"\n\nPrimes beween 7,700 and 8,000 include: "
c = 0
for n = 7700 to 8000
if ( fn IsPrime(n) ) then c += fn IsPrime(n) : printf @"%ld \b", n : count++ : track++
if count = 10 then print : count = 0
next
printf @"There are %ld primes beween 7,700 and 8,000.", track
printf @"\nThe 10,000th prime is: "
c = 0 : n = 1
while ( c < 10000 )
n++
c += fn IsPrime(n)
wend
printf @"%ld", n
end fn
CFTimeInterval t
t = fn CACurrentMediaTime
fn ExtensiblePrimes
printf @"\nCompute time: %.3f ms",(fn CACurrentMediaTime-t)*100
HandleEvents
```

- Output:

The first 20 prime numbers are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 Primes between 100 and 150 include: 101 103 107 109 113 127 131 137 139 149 Primes beween 7,700 and 8,000 include: 7703 7717 7723 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 7927 7933 7937 7949 7951 7963 7993 There are 30 primes beween 7,700 and 8,000. The 10,000th prime is: 104729 Compute time: 73.034 ms

## Go

An implementation of "The Genuine Sieve of Eratosthenese" by Melissa E. O'Niell. This is the paper cited above in the "Faster Alternative Version" of D. The Go example here though strips away optimizations such as a wheel to show the central idea of storing prime multiples in a queue data structure.

```
package main
import (
"container/heap"
"fmt"
)
func main() {
p := newP()
fmt.Print("First twenty: ")
for i := 0; i < 20; i++ {
fmt.Print(p(), " ")
}
fmt.Print("\nBetween 100 and 150: ")
n := p()
for n <= 100 {
n = p()
}
for ; n < 150; n = p() {
fmt.Print(n, " ")
}
for n <= 7700 {
n = p()
}
c := 0
for ; n < 8000; n = p() {
c++
}
fmt.Println("\nNumber beween 7,700 and 8,000:", c)
p = newP()
for i := 1; i < 10000; i++ {
p()
}
fmt.Println("10,000th prime:", p())
}
func newP() func() int {
n := 1
var pq pQueue
top := &pMult{2, 4, 0}
return func() int {
for {
n++
if n < top.pMult { // n is a new prime
heap.Push(&pq, &pMult{prime: n, pMult: n * n})
top = pq[0]
return n
}
// n was next on the queue, it's a composite
for top.pMult == n {
top.pMult += top.prime
heap.Fix(&pq, 0)
top = pq[0]
}
}
}
}
type pMult struct {
prime int
pMult int
index int
}
type pQueue []*pMult
func (q pQueue) Len() int { return len(q) }
func (q pQueue) Less(i, j int) bool { return q[i].pMult < q[j].pMult }
func (q pQueue) Swap(i, j int) {
q[i], q[j] = q[j], q[i]
q[i].index = i
q[j].index = j
}
func (p *pQueue) Push(x interface{}) {
q := *p
e := x.(*pMult)
e.index = len(q)
*p = append(q, e)
}
func (p *pQueue) Pop() interface{} {
q := *p
last := len(q) - 1
e := q[last]
*p = q[:last]
return e
}
```

- Output:

First twenty: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 Between 100 and 150: 101 103 107 109 113 127 131 137 139 149 Number beween 7,700 and 8,000: 30 10,000th prime: 104729

An alternative showing how to use a good and *very* fast open source Sieve of Atkin implementation
via github.com/jbarham/primegen.go.
Due to how Go's imports work, the bellow can be given directly to "`go run`

" or "`go build`

" and the latest version of the primegen package will be fetched and built if it's not already present on the system.
(This example may not be exactly within the scope of this task, but it's a trivial to use and extremely fast prime generator probably worth considering whenever primes are needed in Go.)

```
package main
import (
"fmt"
"github.com/jbarham/primegen.go"
)
func main() {
p := primegen.New()
fmt.Print("First twenty: ")
for i := 0; i < 20; i++ {
fmt.Print(p.Next(), " ")
}
fmt.Print("\nBetween 100 and 150: ")
p.SkipTo(100)
for n := p.Next(); n < 150; n = p.Next() {
fmt.Print(n, " ")
}
p.SkipTo(7700)
fmt.Println("\nNumber beween 7,700 and 8,000:", p.Count(8000))
p.Reset()
for i := 1; i < 1e4; i++ {
p.Next()
}
fmt.Println("10,000th prime:", p.Next())
}
```

## Haskell

This program uses the primes package, which uses a lazy wheel sieve to produce an infinite list of primes.

```
#!/usr/bin/env runghc
import Data.List
import Data.Numbers.Primes
import System.IO
firstNPrimes :: Integer -> [Integer]
firstNPrimes n = genericTake n primes
primesBetweenInclusive :: Integer -> Integer -> [Integer]
primesBetweenInclusive lo hi =
dropWhile (< lo) $ takeWhile (<= hi) primes
nthPrime :: Integer -> Integer
nthPrime n = genericIndex primes (n - 1) -- beware 0-based indexing
main = do
hSetBuffering stdout NoBuffering
putStr "First 20 primes: "
print $ firstNPrimes 20
putStr "Primes between 100 and 150: "
print $ primesBetweenInclusive 100 150
putStr "Number of primes between 7700 and 8000: "
print $ genericLength $ primesBetweenInclusive 7700 8000
putStr "The 10000th prime: "
print $ nthPrime 10000
```

- Output:

First 20 primes: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71] Primes between 100 and 150: [101,103,107,109,113,127,131,137,139,149] Number of primes between 7700 and 8000: 30 The 10000th prime: 104729

### List based

Using list based unbounded sieve from here (runs instantly):

```
λ> take 20 primesW
[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71]
λ> takeWhile (< 150) . dropWhile (< 100) $ primesW
[101,103,107,109,113,127,131,137,139,149]
λ> length . takeWhile (< 8000) . dropWhile (< 7700) $ primesW
30
λ> (!! (10000-1)) primesW
104729
```

### Using analytic formula for primes

There are analytic functions to generate primes. One of such formula is the following:

- p(
*n*) = [2*n*(2*n*+1) {(2*n*-1)!*C*}]

here [*x*] is the integral part of *x*, {*x*} is the fractional part of *x*, and *C* is a real constant (similar to Mill's constant). We can prove that there is a constant *C*, such that p(*n*) is exactly the *n*th prime number for any natural number *n*. The first digits of *C* are *C*=0.359344964622775339841352348439200241924659634...
Haskell have a library of constructive real numbers, where real numbers can be of an arbitrary precision. We can define this constant *C* **exactly** and use the above formula to calculate the *n*th prime. This is not the fastest method, but one of the coolest. And it is not as bad as you may think. It took about 0.3 seconds to calculate 10,000th prime using this formula.

```
{-# LANGUAGE PostfixOperators #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE FlexibleInstances #-}
import Data.Numbers.Primes
import Data.Array.Unboxed hiding ((!))
import qualified Data.Array.Unboxed as Array
import Data.CReal
import Data.CReal.Internal
import GHC.TypeLits
instance KnownNat n => Enum (CReal n) where
toEnum i = fromIntegral i
fromEnum _ = error "Cannot fromEnum CReal"
enumFrom = iterate (+ 1)
enumFromTo n e = takeWhile (<= e) $ iterate (+ 1)n
enumFromThen n m = iterate (+(m-n)) n
enumFromThenTo n m e = if m >= n then takeWhile (<= e) $ iterate (+(m-n)) n
else takeWhile (>= e) $ iterate (+(m-n)) n
-- partial_sum x y a b = (p,q) where
-- p/q = sum_{a<i<=b} x(i) / poduct_{a<j<=j} y(j)
-- The complexity of partial_sum x y 0 n is O(n log n)
partial_sum x y = pq where
pq a b = if a>=b then (0,1)
else if a==b-1 then (fromIntegral $ x b, fromIntegral $ y b )
else (p_ab,q_ab)
where
c=(a+b) `div` 2
(p_ac,q_ac) = pq a c
(p_cb,q_cb) = pq c b
p_ab = p_cb + q_cb*p_ac
q_ab = q_ac*q_cb
-- c is the real constant that is used in the formula for primes
-- c = sum_{1<i} p_i / (2i+1)!
-- where p_i is i-th prime.
-- This will work for any sequence of integers p, where |p_n| < 2n(2n+1) * 0.375
c = crMemoize f where
f n = 2^n * p `div` q where
n' = fromIntegral n
u = head [ceiling (x) | x<-[(n' * log 2/ (log n'-1)/2 ) ..] , 2*x*log (2*x) - 2*x > n'*log 2]
-- Invariant: (2u+1)! > 2^n
ar :: UArray Int Int
ar = listArray (1,u) $ primes
(p,q) = partial_sum (ar Array.!) (\n-> 2*n*(2*n+1) ) 0 u
-- Fractorial part of x
-- By definition it is in the interval [-0.5; 0.5]
-- But it gurantes to work corectly if fractional part of x is in (-0.375; 0.375)
fract x = x - fromIntegral (round (x :: CReal 3))
-- Factorial.
-- The complexity of (n!) is O(n log n) (which is better than O(n^2) for product [1..n] )
(!) :: (RealFrac a, Num b) => a -> b
(!) = fromIntegral . snd . partial_sum (const 0) id 0 . round
-- Analytic function for n-th prime.
-- NB. Strictly speaking this function is not analytic, because it uses factorial, fractional part and round functions
-- To make it truly analytic you need to replace
-- fract x = acos (cos (2*pi*x)) / (2*pi)
-- round x = x - fract x
-- and use the Gamma function instead of factorial.
-- Then you will get analytic function prime :: CReal 0 -> CReal 0
prime n = round( 2*n*(2*n+1) * fract ( c * ((2*n-1)!)))
```

Then you can use this function:

```
λ> :set +s
λ> prime 10000
104729
(0.32 secs, 179,899,272 bytes)
λ> length $ dropWhile (< 7700) $ takeWhile (< 8000) $ map prime [1..]
30
(3.09 secs, 3,418,225,920 bytes)
λ> dropWhile (< 100) $ takeWhile (< 150) $ map prime [1..]
[101,103,107,109,113,127,131,137,139,149]
(0.02 secs, 20,239,464 bytes)
λ> map prime [1..20]
[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71]
(0.01 secs, 10,485,208 bytes)
```

## Icon and Unicon

Only works in Unicon (use of the Heap class). Brute force.

The expression:

```
![2,3,5,7] | (nc := 11) | (nc +:= |wheel2345)
```

is an open-ended sequence generating potential primes.

```
import Collections # to get the Heap class for use as a Priority Queue
record filter(composite, prime) # next composite involving this prime
procedure main()
every writes((primes()\20)||" " | "\n")
every p := primes() do if 100 < p < 150 then writes(p," ") else if p >= 150 then break write()
every (n := 0, p := primes()) do if 7700 < p < 8000 then n +:= 1 else if p >= 8000 then break write(n)
every (i := 1, p := primes()) do if (i+:=1) >= 10000 then break write(p)
end
procedure primes()
local wheel2357, nc
wheel2357 := [2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2,
6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6,
2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10]
suspend sieve(Heap(,getCompositeField), ![2,3,5.7] | (nc := 11) | (nc +:= |!wheel2357))
end
procedure sieve(pQueue, candidate)
local nc
if 0 = pQueue.size() then { # 2 is prime
pQueue.add(filter(candidate*candidate, candidate))
return candidate
}
while candidate > (nc := pQueue.get()).composite do {
nc.composite +:= nc.prime
pQueue.add(nc)
}
pQueue.add(filter(nc.composite+nc.prime, nc.prime))
if candidate < nc.composite then { # new prime found!
pQueue.add(filter(candidate*candidate, candidate))
return candidate
}
end
# Provide a function for comparing filters in the priority queue...
procedure getCompositeField(x); return x.composite; end
```

- Output:

->ePrimes 2 3 5.7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 101 103 107 109 113 127 131 137 139 149 30 104729 ->

## J

Using the p: builtin, http://www.jsoftware.com/help/dictionary/dpco.htm reports "Currently, arguments larger than 2^31 are tested to be prime according to a probabilistic algorithm (Miller-Rabin)".

```
p:i.20
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
(#~ >:&100)i.&.(p:inv) 150
101 103 107 109 113 127 131 137 139 149
#(#~ >:&7700)i.&.(p:inv) 8000
30
p:10000-1
104729
```

Note: p: gives the nth prime, where 0 is first, 1 is second, 2 (cardinal) is third (ordinal) and so on...

Note: 4&p: gives the next prime

```
4 p: 104729
104743
```

## Java

Based on my second C++ solution, which in turn was based on the C solution.

```
import java.util.*;
public class PrimeGenerator {
private int limit_;
private int index_ = 0;
private int increment_;
private int count_ = 0;
private List<Integer> primes_ = new ArrayList<>();
private BitSet sieve_ = new BitSet();
private int sieveLimit_ = 0;
public PrimeGenerator(int initialLimit, int increment) {
limit_ = nextOddNumber(initialLimit);
increment_ = increment;
primes_.add(2);
findPrimes(3);
}
public int nextPrime() {
if (index_ == primes_.size()) {
if (Integer.MAX_VALUE - increment_ < limit_)
return 0;
int start = limit_ + 2;
limit_ = nextOddNumber(limit_ + increment_);
primes_.clear();
findPrimes(start);
}
++count_;
return primes_.get(index_++);
}
public int count() {
return count_;
}
private void findPrimes(int start) {
index_ = 0;
int newLimit = sqrt(limit_);
for (int p = 3; p * p <= newLimit; p += 2) {
if (sieve_.get(p/2 - 1))
continue;
int q = p * Math.max(p, nextOddNumber((sieveLimit_ + p - 1)/p));
for (; q <= newLimit; q += 2*p)
sieve_.set(q/2 - 1, true);
}
sieveLimit_ = newLimit;
int count = (limit_ - start)/2 + 1;
BitSet composite = new BitSet(count);
for (int p = 3; p <= newLimit; p += 2) {
if (sieve_.get(p/2 - 1))
continue;
int q = p * Math.max(p, nextOddNumber((start + p - 1)/p)) - start;
q /= 2;
for (; q >= 0 && q < count; q += p)
composite.set(q, true);
}
for (int p = 0; p < count; ++p) {
if (!composite.get(p))
primes_.add(p * 2 + start);
}
}
private static int sqrt(int n) {
return nextOddNumber((int)Math.sqrt(n));
}
private static int nextOddNumber(int n) {
return 1 + 2 * (n/2);
}
public static void main(String[] args) {
PrimeGenerator pgen = new PrimeGenerator(20, 200000);
System.out.println("First 20 primes:");
for (int i = 0; i < 20; ++i) {
if (i > 0)
System.out.print(", ");
System.out.print(pgen.nextPrime());
}
System.out.println();
System.out.println("Primes between 100 and 150:");
for (int i = 0; ; ) {
int prime = pgen.nextPrime();
if (prime > 150)
break;
if (prime >= 100) {
if (i++ != 0)
System.out.print(", ");
System.out.print(prime);
}
}
System.out.println();
int count = 0;
for (;;) {
int prime = pgen.nextPrime();
if (prime > 8000)
break;
if (prime >= 7700)
++count;
}
System.out.println("Number of primes between 7700 and 8000: " + count);
int n = 10000;
for (;;) {
int prime = pgen.nextPrime();
if (prime == 0) {
System.out.println("Can't generate any more primes.");
break;
}
if (pgen.count() == n) {
System.out.println(n + "th prime: " + prime);
n *= 10;
}
}
}
}
```

- Output:

First 20 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 Primes between 100 and 150: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149 Number of primes between 7700 and 8000: 30 10000th prime: 104729 100000th prime: 1299709 1000000th prime: 15485863 10000000th prime: 179424673 100000000th prime: 2038074743 Can't generate any more primes.

## JavaScript

**primeGenerator(num, showPrimes)**
This function takes two arguments:

**num** is either an integer as a limit, or an array of two integers to present a range;

**showPrimes** is a boolean to indicate whether the result should be a list (if *true*) or a single number (if *false*).

Sounds a bit weird, but I hope it will be intelligible by the testing examples below. First the code:

```
function primeGenerator(num, showPrimes) {
var i,
arr = [];
function isPrime(num) {
// try primes <= 16
if (num <= 16) return (
num == 2 || num == 3 || num == 5 || num == 7 || num == 11 || num == 13
);
// cull multiples of 2, 3, 5 or 7
if (num % 2 == 0 || num % 3 == 0 || num % 5 == 0 || num % 7 == 0)
return false;
// cull square numbers ending in 1, 3, 7 or 9
for (var i = 10; i * i <= num; i += 10) {
if (num % (i + 1) == 0) return false;
if (num % (i + 3) == 0) return false;
if (num % (i + 7) == 0) return false;
if (num % (i + 9) == 0) return false;
}
return true;
}
if (typeof num == "number") {
for (i = 0; arr.length < num; i++) if (isPrime(i)) arr.push(i);
// first x primes
if (showPrimes) return arr;
// xth prime
else return arr.pop();
}
if (Array.isArray(num)) {
for (i = num[0]; i <= num[1]; i++) if (isPrime(i)) arr.push(i);
// primes between x .. y
if (showPrimes) return arr;
// number of primes between x .. y
else return arr.length;
}
// throw a default error if nothing returned yet
// (surrogate for a quite long and detailed try-catch-block anywhere before)
throw("Invalid arguments for primeGenerator()");
}
```

**Test**

```
// first 20 primes
console.log(primeGenerator(20, true));
// primes between 100 and 150
console.log(primeGenerator([100, 150], true));
// numbers of primes between 7700 and 8000
console.log(primeGenerator([7700, 8000], false));
// the 10,000th prime
console.log(primeGenerator(10000, false));
```

**Output**

```
Array [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 59, 61, 67, 71 ]
Array [ 101, 103, 107, 109, 113, 127, 131, 137, 139, 149 ]
30
104729
```

## jq

Recent versions of jq include extensive support for unbounded stream generators, but in this section, we present a solution to the tasks that should work with any version of jq from 1.4 onwards. That is, instead of using an unbounded generator of a stream of primes, the core of the approach adopted here is a function, "extend_primes", which can be applied recursively to generate arbitrarily many, or indefinitely many, primes, as illustrated by the function named "primes" below.

**Preliminaries:**

```
# Recent versions of jq include the following definition:
# until/2 loops until cond is satisfied,
# and emits the value satisfying the condition:
def until(cond; next):
def _until:
if cond then . else (next|_until) end;
_until;
def count(cond): reduce .[] as $x (0; if $x|cond then .+1 else . end);
```

**Prime numbers:**

```
# Is the input integer a prime?
# "previous" must be the array of sorted primes greater than 1 up to (.|sqrt)
def is_prime(previous):
. as $in
| (previous|length) as $plength
| [false, 0] # state: [found, ix]
| until( .[0] or .[1] >= $plength;
[ ($in % previous[.[1]]) == 0, .[1] + 1] )
| .[0] | not ;
# extend_primes expects its input to be an array consisting of
# previously found primes, in order, and extends that array:
def extend_primes:
if . == null or length == 0 then [2]
else . as $previous
| if . == [2] then [2,3]
else . + [(2 + .[length-1]) | until( is_prime($previous) ; . + 2)]
end
end;
# If . is an integer > 0 then produce an array of . primes;
# otherwise emit an unbounded stream of primes:
def primes:
. as $n
| if type == "number" and $n > 0 then
null | until( length == $n; extend_primes )
else [2] | recurse(extend_primes) | .[length - 1]
end;
# Primes up to and possibly including n:
def primes_upto(n):
until( .[length-1] > n; extend_primes )
| if .[length-1] > n then .[0:length-1] else . end;
```

**The tasks:**
The tasks are completed separately here to highlight the fact that by using "extend_primes", each task can be readily completed without generating unnecessarily many primes.

```
"First 20 primes:", (20 | primes), "",
"Primes between 100 and 150:",
(primes_upto(150) | map(select( 100 < .))), "",
"The 10,000th prime is \( 10000 | primes | .[length - 1] )", "",
(( primes_upto(8000) | count( . > 7700) | length) as $length
| "There are \($length) primes twixt 7700 and 8000.")
```

- Output:

```
$ jq -r -c -n -f Extensible_prime_generator.jq
First 20 primes:
[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71]
Primes between 100 and 150:
[101,103,107,109,113,127,131,137,139,149]
The 10,000th prime is 104729
There are 30 primes twixt 7700 and 8000.
```

## Julia

Julia's Primes package, included in the distribution, is exact up to 2^64 = 18446744073709551616. After that, Primes can use the BigInt data type, and then may use a probabalistic prime determination algorithm for such integers of arbitrarily large size. The probabilistic formula is tune-able, and by default determines primes with Knuth's recommended level for cryptography of an error less than (0.25)^25 = 8.881784197001252e-16, or 1 in 1125899906842624.

```
using Primes
sum = 2
currentprime = 2
for i in 2:100000
currentprime = nextprime(currentprime + 1)
sum += currentprime
end
println("The sum of the first 100,000 primes is $sum")
curprime = 1
arr = zeros(Int, 20)
for i in 1:20
curprime = nextprime(curprime + 1)
arr[i] = curprime
end
println("The first 20 primes are ", arr)
println("the primes between 100 and 150 are ", primes(100,150))
println("The number of primes between 7,700 and 8,000 is ", length(primes(7700, 8000)))
println("The 10,000th prime is ", prime(10000))
```

- Output:

The sum of the first 100,000 primes is 62260698721 The first 20 primes are [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71] the primes between 100 and 150 are [101, 103, 107, 109, 113, 127, 131, 137, 139, 149] The number of primes between 7,700 and 8,000 is 30 The 10,000th prime is 104729

**Alternative True Generator**

The above code just uses the "Primes" package as a set of tools to solve the tasks. The following code creates a very simple generator using `isprime` inside a iterator and then uses that to solve the tasks:

```
using Primes: isprime
PrimesGen() = Iterators.filter(isprime, Iterators.countfrom(Int64(2)))
print("Sum of first 100,000 primes: ")
println(Iterators.sum(Iterators.take(PrimesGen(), 100000)))
print("First 20 primes: ( ")
foreach((p->print(p," ")), Iterators.take(PrimesGen(), 20))
println(")")
print("Primes between 100 and 150: ( ")
for p in Iterators.filter((p->p>=100), PrimesGen()) p > 150 && break; print(p, " ") end
println(")")
let cnt = 0
for p in PrimesGen()
p > 8000 && break; if p > 7700 cnt += 1 end
end; println("Number of primes between 7700 and 8000: ", cnt)
end
println("The 10,000th prime: ", Iterators.first(Iterators.drop(PrimesGen(), 9999)))
println()
```

This outputs:

- Output:

Sum of first 100,000 primes: 62260698721 First 20 primes: ( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ) Primes between 100 and 150: ( 101 103 107 109 113 127 131 137 139 149 ) Number of primes between 7700 and 8000: 30 The 10,000th prime: 104729

To show it's speed, lets use it to solve the Euler Problem 10 of calculating the sum of the primes to two million as follows:

```
using Printf: @printf
@time let sm = 0
for p in Iterators.filter(isprime, Iterators.countfrom(UInt64(2)))
p > 2000000 && break
sm += p
end; @printf("%d\n", sm) end
```

which outputs:

- Output:

142913828922 0.783845 seconds (328.02 k allocations: 5.042 MiB)

As shown, this is of adequate speed for this smallish range; however it wouldn't be adequate to do the same for a range of two billion.

This is an "infinite" iterator whose range is limited by the size of `Int64`, but as it will take about 300 thousand years to get there, it isn't much of a concern.

**An "infinite" iterator based on a bit-packed page-segmented Sieve of Eratosthenes**

The above code is more than adequate to solve the trivial tasks as required here, but is really too slow for "industrial strength" tasks for ranges of billions. The following code uses the Page Segmented Algorithm from Sieve_of_Eratosthenes#Julia to solve the task:

```
using Printf: @printf
print("Sum of first 100,000 primes: ")
println(Iterators.sum(Iterators.take(PrimesPaged(), 100000)))
print("First 20 primes: ( ")
foreach((p->@printf("%d ", p)), Iterators.take(PrimesPaged(), 20))
println(")")
print("Primes between 100 and 150: ( ")
for p in Iterators.filter((p->p>=100), PrimesPaged()) p > 150 && break; @printf("%d ", p)) end
println(")")
let cnt = 0
for p in PrimesPaged()
p > 8000 && break; if p > 7700 cnt += 1 end
end; println("Number of primes between 7700 and 8000: ", cnt)
end
@printf("The 10,000th prime: %d\n", Iterators.first(Iterators.drop(PrimesPaged(), 9999)))
```

to produce the same output much faster.

To show how much faster it is, doing the same Euler Problem 10 as follows:

```
using Printf: @printf
@time let sm = 0
for p in PrimesPaged()
p > 2000000 && break
sm += p
end; @printf("%d\n", sm) end
```

produces:

- Output:

142913828922 0.016826 seconds (60 allocations: 23.891 KiB)

showing it is as over 40 times faster, but it will definitely get even relatively faster with increasing range as this range is relatively trivial for page segmentation and there are more optimizations one can make.

This generator is also "infinite" to the `UInt64` range, but now will "only" take hundreds of years to get there.

## Kotlin

Although we could use the java.math.BigInteger type to generate arbitrarily large primes, there is no need to do so here as the primes to be generated are well within the limits of the 4-byte Int type. ((workwith|Kotlin|version 1.3}}

```
fun isPrime(n: Int) : Boolean {
if (n < 2) return false
if (n % 2 == 0) return n == 2
if (n % 3 == 0) return n == 3
var d : Int = 5
while (d * d <= n) {
if (n % d == 0) return false
d += 2
if (n % d == 0) return false
d += 4
}
return true
}
fun generatePrimes() = sequence {
yield(2)
var p = 3
while (p <= Int.MAX_VALUE) {
if (isPrime(p)) yield(p)
p += 2
}
}
fun main(args: Array<String>) {
val primes = generatePrimes().take(10000) // generate first 10,000 primes
println("First 20 primes : ${primes.take(20).toList()}")
println("Primes between 100 and 150 : ${primes.filter { it in 100..150 }.toList()}")
println("Number of primes between 7700 and 8000 = ${primes.filter { it in 7700..8000 }.count()}")
println("10,000th prime = ${primes.last()}")
}
```

- Output:

First 20 primes : [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71] Primes between 100 and 150 : [101, 103, 107, 109, 113, 127, 131, 137, 139, 149] Number of primes between 7700 and 8000 = 30 10,000th prime = 104729

**Alternate with better performance**

Of all the current submissions on the page, the above code has got to be the worst. While it is adequate to solve the trivial tasks required by the page, it is a Trial Division Sieve and has O(n^3/2)) asymptotic performance over `n`, the range, where even a purely functional incremental sieve has O(n log n) performance. There is no need to consider BigInteger at all for a prime generator starting from the lowest to the Long number range, as it will never get there in less than 100's of years. Even though the above code implements a rudimentary wheel factorization, it will still be extremely slow as that only provides constant factor gains.

The following odds-only incremental Sieve of Eratosthenes generator has O(n log (log n)) performance due to using a (mutable) HashMap:

```
fun primesHM(): Sequence<Int> = sequence {
yield(2)
fun oddprms(): Sequence<Int> = sequence {
yield(3); yield(5) // need at least 2 for initialization
val hm = HashMap<Int,Int>()
hm.put(9, 6)
val bps = oddprms().iterator(); bps.next(); bps.next() // skip past 5
yieldAll(generateSequence(SieveState(7, 5, 25)) {
ss ->
var n = ss.n; var q = ss.q
n += 2
while ( n >= q || hm.containsKey(n)) {
if (n >= q) {
val inc = ss.bp shl 1
hm.put(n + inc, inc)
val bp = bps.next(); ss.bp = bp; q = bp * bp
}
else {
val inc = hm.remove(n)!!
var next = n + inc
while (hm.containsKey(next)) {
next += inc
}
hm.put(next, inc)
}
n += 2
}
ss.n = n; ss.q = q
ss
}.map { it.n })
}
yieldAll(oddprms())
}
```

it is faster than the first example even though not using wheel factorization (other than odds-only) and rapidly pulls far ahead of it with increasing range such that it is usable to a range of 100 million in the order of 10 seconds.

**Alternate with "industrial strength" performance**

For ranges of a billion and more, one needs a sieve based on Page Segmented mutable arrays. The last code on the Sieve of Eratosthenes Task page at: Sieve_of_Eratosthenes#Unbounded_Versions_2 can do the job. When called with the following same `main` as the first example with `primesPaged()` substituted for `generatePrimes()` or `primesHM()`, it produces the same output.

It can count the primes to one billion in about 15 seconds on a slow tablet CPU (Intel x5-Z8350 at 1.92 Gigahertz) with the following code:

```
primesPaged().takeWhile { it <= 1_000_000_000 }.count()
```

Further speed-ups can be achieved of about a factor of four with maximum wheel factorization and by multi-threading by the factor of the effective number of cores used, but there is little point when most of the execution time as a generator is spend iterating over the results.

In order to take advantage of those optimizations, one needs to write functions that work directly with the provided sequence of culled bit pages such as the provided `countPrimesTo` function does, which counts the primes without the iteration about three times as fast.

## Lua

The modest requirements of this task allow for naive implementations, such as what follows. This generator does not even use its own list of primes to help determine subsequent primes! (though easily fixed) So, it it sufficient, but not efficient.

```
local primegen = {
count_limit = 2,
value_limit = 3,
primelist = { 2, 3 },
nextgenvalue = 5,
nextgendelta = 2,
tbd = function(n)
if n < 2 then return false end
if n % 2 == 0 then return n==2 end
if n % 3 == 0 then return n==3 end
local limit = math.sqrt(n)
for f = 5, limit, 6 do
if n % f == 0 or n % (f+2) == 0 then return false end
end
return true
end,
needmore = function(self)
return (self.count_limit ~= nil and #self.primelist < self.count_limit)
or (self.value_limit ~= nil and self.nextgenvalue < self.value_limit)
end,
generate = function(self, count_limit, value_limit)
self.count_limit = count_limit
self.value_limit = value_limit
while self:needmore() do
if (self.tbd(self.nextgenvalue)) then
self.primelist[#self.primelist+1] = self.nextgenvalue
end
self.nextgenvalue = self.nextgenvalue + self.nextgendelta
self.nextgendelta = 6 - self.nextgendelta
end
end,
filter = function(self, f)
local list = {}
for k,v in ipairs(self.primelist) do
if (f(v)) then list[#list+1] = v end
end
return list
end,
}
primegen:generate(20, nil)
print("First 20 primes: " .. table.concat(primegen.primelist, ", "))
primegen:generate(nil, 150)
print("Primes between 100 and 150: " .. table.concat(primegen:filter(function(v) return v>=100 and v<=150 end), ", "))
primegen:generate(nil, 8000)
print("Number of primes between 7700 and 8000: " .. #primegen:filter(function(v) return v>=7700 and v<=8000 end))
primegen:generate(10000, nil)
print("The 10,000th prime: " .. primegen.primelist[#primegen.primelist])
primegen:generate(100000, nil)
print("The 100,000th prime: " .. primegen.primelist[#primegen.primelist])
```

- Output:

First 20 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 Primes between 100 and 150: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149 Number of primes between 7700 and 8000: 30 The 10,000th prime: 104729 The 100,000th prime: 1299709

## Lingo

The following script implements a Sieve of Eratosthenes that is automatically extended when a method call needs a higher upper limit.

```
-- parent script "sieve"
property _sieve
----------------------------------------
-- @constructor
----------------------------------------
on new (me)
me._sieve = []
me._primeSieve(100) -- arbitrary initial size of sieve
return me
end
----------------------------------------
-- Returns sorted list of first n primes p with p >= a (default: a=1)
----------------------------------------
on getNPrimes (me, n, a)
if voidP(a) then a = 1
i = a
res = []
repeat while TRUE
if i>me._sieve.count then me._primeSieve(2*i)
if me._sieve[i] then res.add(i)
if res.count=n then return res
i = i +1
end repeat
end
----------------------------------------
-- Returns sorted list of primes p with a <= p <= b
----------------------------------------
on getPrimesInRange (me, a, b)
if me._sieve.count<b then me._primeSieve(b)
primes = []
repeat with i = a to b
if me._sieve[i] then primes.add(i)
end repeat
return primes
end
----------------------------------------
-- Returns nth prime
----------------------------------------
on getNthPrime (me, n)
if me._sieve.count<2*n then me._primeSieve(2*n)
i = 0
found = 0
repeat while TRUE
i = i +1
if i>me._sieve.count then me._primeSieve(2*i)
if me._sieve[i] then found=found+1
if found=n then return i
end repeat
end
----------------------------------------
-- Sieve of Eratosthenes
----------------------------------------
on _primeSieve (me, limit)
if me._sieve.count>=limit then
return
else if me._sieve.count>0 then
return me._complementSieve(limit)
end if
me._sieve = [0]
repeat with i = 2 to limit
me._sieve[i] = 1
end repeat
c = sqrt(limit)
repeat with i = 2 to c
if (me._sieve[i]=0) then next repeat
j = i*i
repeat while (j<=limit)
me._sieve[j] = 0
j = j + i
end repeat
end repeat
end
----------------------------------------
-- Expands existing sieve to new limit
----------------------------------------
on _complementSieve (me, n)
n1 = me._sieve.count
repeat with i = n1+1 to n
me._sieve[i] = 1
end repeat
c1 = sqrt(n1)
repeat with i = 2 to c1
if (me._sieve[i]=0) then next repeat
j = n1 - (n1 mod i)
repeat while (j<=n)
me._sieve[j] = 0
j = j + i
end repeat
end repeat
c = sqrt(n)
repeat with i = c1+1 to c
if (me._sieve[i]=0) then next repeat
j = i*i
repeat while (j<=n)
me._sieve[j] = 0
j = j + i
end repeat
end repeat
end
```

```
sieve = script("sieve").new()
put "First twenty primes: " & sieve.getNPrimes(20)
put "Primes between 100 and 150: "& sieve.getPrimesInRange(100, 150)
put "Number of primes between 7,700 and 8,000: " & sieve.getPrimesInRange(7700, 8000).count
put "The 10,000th prime: " & sieve.getNthPrime(10000)
```

- Output:

-- "First twenty primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]" -- "Primes between 100 and 150: [101, 103, 107, 109, 113, 127, 131, 137, 139, 149]" -- "Number of primes between 7,700 and 8,000: 30" -- "The 10,000th prime: 104729"

## M2000 Interpreter

The fancy way, using two lambda which shares closures (they are pointers, so lanbdas get copies of pointers, which is by value passing, and by reference too). Also I use my loved GOTO. Change 200th to 10000th and wait... A 5@ is a literal for Decimals (Variables are Variant types, but when they get a value they hold that value type. Array items or in other containers they, get whatever we want, anytime)

Version 2 Now we can make a bigger computation using Fast! mode which eliminate Gui/console refresh to gain speed. We can make a refresh each 50primes, so we have a tiny delay on refreshing if we move a window above M2000 console.

Also I change IsPrime to not add to Inventory Known1, because we want this inventory to have all primes without any missing until the last one.

I provide another IsPrime2 which use PrimeNth to add to Known1.

Inventories are reference type. Lambda functions are value type, but closures which are reference type copied the reference by value, so we get then by reference. Inventories start now with 3 known primes, and PrimeNth works for odd numbers only (see x+=2 before the loop statement)

Loop statement check a flag in a block of code ({ }), so when the block ends restart again (resetting the loop flag)

```
Module CheckPrimes {
\\ Inventories are lists, Known and Known1 are pointers to Inventories
Inventory Known=1:=2@,2:=3@,3:=5@
Inventory Known1=2@, 3@, 5@
\\ In a lambda all closures are copies
\\ but Known and Know1 are copies of pointers
\\ so are closures like by reference
PrimeNth=lambda Known, Known1 (n as long) -> {
if n<1 then Error "Only >=1"
if exist(known, n) then =eval(known) : exit
if n>5 then {
i=len(known1)
x=eval(known1, i-1)+2
} else x=5 : i=2
{
if i=n then =known(n) : exit
ok=false
if frac(x) then 1000
if frac(x/2) else 1000
if frac(x/3) else 1000
x1=sqrt(x) : d=5@
Repeat
if frac(x/d ) else exit
d += 2: if d>x1 then ok=true : exit
if frac(x/d) else exit
d += 4: if d<= x1 else ok=true: exit
Always
1000 If ok then i++:Append Known, i:=x : if not exist(Known1, x) then Append Known1, x
x+=2 : Loop }
}
\\ IsPrime has same closure, Known1
IsPrime=lambda Known1 (x as decimal) -> {
if exist(Known1, x) then =true : exit
if Eval(Known1, len(Known1)-1)>x then exit
if frac(x/2) else exit
if frac(x/3) else exit
x1=sqrt(x):d = 5@
{if frac(x/d ) else exit
d += 2: if d>x1 then =true : exit
if frac(x/d) else exit
d += 4: if d<= x1 else =true: exit
loop
}
}
\\ fill Known1, PrimeNth is a closure here
IsPrime2=lambda Known1, PrimeNth (x as decimal) -> {
if exist(Known1, x) then =true : exit
i=len(Known1)
if Eval(Known1, i-1)>x then exit
{
z=PrimeNth(i)
if z<x then loop else.if z=x then =true :exit
i++
}
}
Print "First twenty primes"
n=PrimeNth(20)
For i=1 to 20 : Print Known(i),: Next i
Print
Print "Primes between 100 and 150:"
c=0
For i=100 to 150
If IsPrime2(i) Then print i, : c++
Next i
Print
Print "Count:", c
Print "Primes between 7700 and 8000:"
c=0
For i=7700 to 8000
If IsPrime(i) Then print i, : c++
Next i
Print
Print "Count:", c
Print "200th Prime:"
Print PrimeNth(200)
Print "List from 190th to 199th Prime:"
For i=190 to 199 : Print Known(i), : Next i
Print
Print "Wait"
Refresh ' because refresh happen on next Print, which take time
' using set fast! we get no respond from GUI/M2000 Console
' also Esc, Break and Ctrl+C not work
' we have to use Refresh each 500 primes to have one Refresh
Set fast !
for i=500 to 10000 step 50: m=PrimeNth(i): Print "."; :Refresh:Next i
Print
Print "10000th Prime:", PrimeNth(10000)
' reset speed to fast (there are three levels: slow/fast/fast!)
set fast
Print
Rem 1 : Print Known
Rem 2: Print Known1
}
CheckPrimes
```

- Output:

First twenty primes 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 Primes between 100 and 150: 101 103 107 109 113 127 131 137 139 149 Count: 10 Primes between 7700 and 8000: 7703 7717 7723 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 7927 7933 7937 7949 7951 7963 7993 Count: 30 200th Prime: 1223 List from 190th to 199th Prime: 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 Wait .... (truncate for output) 10000th Prime: 104729

### Code Optimization

We can drop ok variable from PrimeNth, using a second label. Statement Restart, restart the block. Labels are hashed when first time used.

I use same indentation so you can copy it at same position as in example above. A loop statement mark once the current block for restart after then last statement on block.

```
PrimeNth=lambda Known, Known1 (n as long) -> {
if n<1 then Error "Only >=1"
if exist(known, n) then =eval(known) : exit
if n>5 then {
i=len(known1)
x=eval(known1, i-1)+2
} else x=5 : i=2
{
if i=n then =known(n) : exit
if frac(x) then 999
if frac(x/2) else 999
if frac(x/3) else 999
x1=sqrt(x) : d=5@
{if frac(x/d ) else 999
d += 2: if d>x1 then 1000
if frac(x/d) else 999
d += 4: if d<= x1 else 1000
loop
}
999 x++ : Restart
1000 i++:Append Known, i:=x : if not exist(Known1, x) then Append Known1, x
x++ : Loop }
}
```

## Mathematica / Wolfram Language

Prime and PrimePi use sparse caching and sieving. For large values, the Lagarias–Miller–Odlyzko algorithm for PrimePi is used, based on asymptotic estimates of the density of primes, and is inverted to give Prime. PrimeQ first tests for divisibility using small primes, then uses the Miller–Rabin strong pseudoprime test base 2 and base 3, and then uses a Lucas test.

```
Prime[Range[20]]
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71}
Select[Range[100,150], PrimeQ]
{101, 103, 107, 109, 113, 127, 131, 137, 139, 149}
PrimePi[8000] - PrimePi[7700]
30
Prime[10000]
104729
```

## Nim

For such trivial ranges as the task requirements or solving the Euler Problem 10 of summing the primes to two million, a basic generator such as the hash table based version from the Sieve_of_Eratosthenes#Nim_Unbounded_Versions section of the Sieve of Eratosthenes task will suffice, as follows:

```
import tables
type PrimeType = int
proc primesHashTable(): iterator(): PrimeType {.closure.} =
iterator output(): PrimeType {.closure.} =
# some initial values to avoid race and reduce initializations...
yield 2.PrimeType; yield 3.PrimeType; yield 5.PrimeType; yield 7.PrimeType
var h = initTable[PrimeType,PrimeType]()
var n = 9.PrimeType
let bps = primesHashTable()
var bp = bps() # advance past 2
bp = bps(); var q = bp * bp # to initialize with 3
while true:
if n >= q:
let inc = bp + bp
h[n + inc] = inc
bp = bps(); q = bp * bp
elif h.hasKey(n):
var inc: PrimeType
discard h.take(n, inc)
var nxt = n + inc
while h.hasKey(nxt): nxt += inc # ensure no duplicates
h[nxt] = inc
else: yield n
n += 2.PrimeType
output
var num = 0
stdout.write "The first 20 primes are: "
var iter = primesHashTable()
for p in iter():
if num >= 20: break else: stdout.write(p, " "); num += 1
echo ""
stdout.write "The primes between 100 and 150 are: "
iter = primesHashTable()
for p in iter():
if p >= 150: break
if p >= 100: stdout.write(p, " ")
echo ""
num = 0
iter = primesHashTable()
for p in iter():
if p > 8000: break
if p >= 7700: num += 1
echo "The number of primes between 7700 and 8000 is: ", num
num = 1
iter = primesHashTable()
for p in iter():
if num >= 10000:
echo "The 10,000th prime is: ", p
break
num += 1
var sum = 0
iter = primesHashTable()
for p in iter():
if p >= 2_000_000:
echo "The sum of the primes to two million is: ", sum
break
sum += p
```

- Output:

The first 20 primes are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 The primes between 100 and 150 are: 101 103 107 109 113 127 131 137 139 149 The number of primes between 7700 and 8000 is: 30 The 10,000th prime is: 104729 The sum of the primes to two million is: 142913828922

The code isn't particularly fast but more than adequate to run this trivial series of tasks and capable for ranges of a few tens of millions. It is limited by the maximum size of the `int` numeric range, but is too slow to ever reach that in a reasonable time, as in many minutes to even reach the 32-bit integer number range of over two billion let alone to reach the integer limit for 64-bit machines.

**Alternate much faster Page Segmented Mutable Bit-Packed Seq version**

The same Sieve of Eratosthenes Nim Unbounded Versions section as above includes a Page Segmented version at the end that can be used the same way for ranges of a billion in only a second (or something approaching a hundred times faster); this can be used for "industrial strength ranges" as in many billions in a reasonable time. Its limit is just the numeric range used, so for 64-bit prime times is about 2e10 in years.

To use that sieve, just substitute the following in the code:

```
for p in primesPaged():
```

wherever the following is in the code:

```
for p in iter():
```

and one doesn't need the lines including `iter` just above these lines at all.

As noted in that section, using an extensible generator isn't the best choice for huge ranges as much higher efficiency can be obtained by using functions that deal directly with the composite culled packed-bit array representations directly as the `countPrimesTo` function does. This makes further improvements to the culling efficiency pointless, as even if one were to reduce the culling speed to zero, it still would take several seconds per range of a billion to enumerate the found primes as a generator.

## OCaml

- 2-3-5-7-wheel postponed incremental sieve

```
module IntMap = Map.Make(Int)
let rec steps =
4 :: 2 :: 4 :: 6 :: 2 :: 6 :: 4 :: 2 :: 4 :: 6 :: 6 :: 2 ::
6 :: 4 :: 2 :: 6 :: 4 :: 6 :: 8 :: 4 :: 2 :: 4 :: 2 :: 4 ::
8 :: 6 :: 4 :: 6 :: 2 :: 4 :: 6 :: 2 :: 6 :: 6 :: 4 :: 2 ::
4 :: 6 :: 2 :: 6 :: 4 :: 2 :: 4 :: 2 :: 10 :: 2 :: 10 :: 2 :: steps
let not_in_wheel =
let scan i =
let rec loop n w = n < 223 && (i = n mod 210
|| match w with [] -> assert false | d :: w' -> loop (n + d) w')
in not (loop 13 steps)
in Array.init 210 scan
let seq_primes =
let rec calc ms m p2 =
if not_in_wheel.(m mod 210) || IntMap.mem m ms
then calc ms (m + p2) p2
else IntMap.add m p2 ms
in
let rec next c p pp ps whl ms () =
match whl with
| [] -> assert false
| d :: w -> match IntMap.min_binding_opt ms with
| Some (m, p2) when c = m ->
next (c + d) p pp ps w (calc (IntMap.remove m ms) (m + p2) p2) ()
| _ when c < pp -> Seq.Cons (c, next (c + d) p pp ps w ms)
| _ -> match ps () with
| Seq.Cons (p', ps') -> let p2' = p + p in
next (c + d) p' (p' * p') ps' w (calc ms (pp + p2') p2') ()
| _ -> assert false
in
let rec ps () = next 13 11 121 ps steps IntMap.empty () in
Seq.cons 2 (Seq.cons 3 (Seq.cons 5 (Seq.cons 7 (Seq.cons 11 ps))))
```

- Test code

```
let seq_show sq =
print_newline (Seq.iter (Printf.printf " %u") sq)
let () =
seq_primes |> Seq.take 20 |> seq_show;
seq_primes |> Seq.drop_while ((>) 100) |> Seq.take_while ((>) 150) |> seq_show;
seq_primes |> Seq.drop_while ((>) 7700) |> Seq.take_while ((>) 8000)
|> Seq.length |> Printf.printf " %u primes\n";
seq_primes |> Seq.drop 9999 |> Seq.take 1 |> seq_show
```

- Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 101 103 107 109 113 127 131 137 139 149 30 primes 104729

## PARI/GP

PARI includes a nice prime generator which is quite extensible:

```
void
showprimes(GEN lower, GEN upper)
{
forprime_t T;
if (!forprime_init(&T, a,b)) return;
while(forprime_next(&T))
{
pari_printf("%Ps\n", T.pp);
}
}
```

Most of these functions are already built into GP:

```
primes(20)
primes([100,150])
#primes([7700,8000]) /* or */
s=0; forprime(p=7700,8000,s++); s
prime(10000)
```

## Pascal

Limited to 2.7e14. Much faster than the other Version First the unit.Still a work in progress.

versus primesieve no chance at all, even single threaded 5x faster :-) ./primesieve -v primesieve 7.2, <https://primesieve.org> Copyright (C) 2010 - 2018 Kim Walisch ./primesieve -t1 100000071680 Sieve size = 256 KiB Threads = 1 100% Seconds: 19.891 Primes: 4118057696

```
unit primsieve;
{$IFDEF FPC}
{$MODE objFPC}{$Optimization ON,ALL}
{$IFEND}
{segmented sieve of Erathostenes using only odd numbers}
{using presieved sieve of small primes, to reduce the most time consuming}
interface
procedure InitPrime;
procedure NextSieve;
function SieveStart:Uint64;
function SieveSize :LongInt;
function Nextprime: Uint64;
function StartCount :Uint64;
function TotalCount :Uint64;
function PosOfPrime: Uint64;
implementation
uses
sysutils;
const
smlPrimes :array [0..10] of Byte = (2,3,5,7,11,13,17,19,23,29,31);
maxPreSievePrimeNum = 7;
maxPreSievePrime = 17;//smlPrimes[maxPreSievePrimeNum];
cSieveSize = 16384 * 4; //<= High(Word)+1 // Level I Data Cache
type
tSievePrim = record
svdeltaPrime:word;//diff between actual and new prime
svSivOfs:word; //Offset in sieve
svSivNum:LongWord;//1 shl (1+16+32) = 5.6e14
end;
tpSievePrim = ^tSievePrim;
var
//sieved with primes 3..maxPreSievePrime.here about 255255 Byte
{$ALIGN 32}
preSieve :array[0..3*5*7*11*13*17-1] of Byte;//must be > cSieveSize
{$ALIGN 32}
Sieve :array[0..cSieveSize-1] of Byte;
{$ALIGN 32}
//prime = FoundPrimesOffset + 2*FoundPrimes[0..FoundPrimesCnt]
FoundPrimes : array[0..cSieveSize] of word;
{$ALIGN 32}
sievePrimes : array[0..78498] of tSievePrim;// 1e6^2 ->1e12
// sievePrimes : array[0..664579] of tSievePrim;// maximum 1e14
FoundPrimesOffset : Uint64;
FoundPrimesCnt,
FoundPrimesIdx,
FoundPrimesTotal,
SieveNum,
SieveMaxIdx,
preSieveOffset,
LastInsertedSievePrime :NativeUInt;
procedure CopyPreSieveInSieve; forward;
procedure CollectPrimes; forward;
procedure sieveOneSieve; forward;
procedure Init0Sieve; forward;
procedure SieveOneBlock; forward;
//****************************************
procedure preSieveInit;
var
i,pr,j,umf : NativeInt;
Begin
fillchar(preSieve[0],SizeOf(preSieve),#1);
i := 1;
pr := 3;// starts with pr = 3
umf := 1;
repeat
IF preSieve[i] =1 then
Begin
pr := 2*i+1;
j := i;
repeat
preSieve[j] := 0;
inc(j,pr);
until j> High(preSieve);
umf := umf*pr;
end;
inc(i);
until (pr = maxPreSievePrime)OR(umf>High(preSieve)) ;
preSieveOffset := 0;
end;
function InsertSievePrimes(PrimPos:NativeInt):NativeInt;
var
delta :NativeInt;
i,pr,loLmt : NativeUInt;
begin
i := 0;
//ignore first primes already sieved with
if SieveNum = 0 then
i := maxPreSievePrimeNum;
pr :=0;
loLmt := Uint64(SieveNum)*(2*cSieveSize);
delta := loLmt-LastInsertedSievePrime;
with sievePrimes[PrimPos] do
Begin
pr := FoundPrimes[i]*2+1;
svdeltaPrime := pr+delta;
delta := pr;
end;
inc(PrimPos);
for i := i+1 to FoundPrimesCnt-1 do
Begin
IF PrimPos > High(sievePrimes) then
BREAK;
with sievePrimes[PrimPos] do
Begin
pr := FoundPrimes[i]*2+1;
svdeltaPrime := (pr-delta);
delta := pr;
end;
inc(PrimPos);
end;
LastInsertedSievePrime := loLmt+pr;
result := PrimPos;
end;
procedure CalcSievePrimOfs(lmt:NativeUint);
//lmt High(sievePrimes)
var
i,pr : NativeUInt;
sq : Uint64;
begin
pr := 0;
i := 0;
repeat
with sievePrimes[i] do
Begin
pr := pr+svdeltaPrime;
IF sqr(pr) < (cSieveSize*2) then
Begin
svSivNum := 0;
svSivOfs := (pr*pr-1) DIV 2;
end
else
Begin
SieveMaxIdx := i;
pr := pr-svdeltaPrime;
BREAK;
end;
end;
inc(i);
until i > lmt;
for i := i to lmt do
begin
with sievePrimes[i] do
Begin
pr := pr+svdeltaPrime;
sq := sqr(pr);
svSivNum := sq DIV (2*cSieveSize);
svSivOfs := ( (sq - Uint64(svSivNum)*(2*cSieveSize))-1)DIV 2;
end;
end;
end;
procedure sievePrimesInit;
var
i,j,pr,PrimPos:NativeInt;
Begin
LastInsertedSievePrime := 0;
preSieveOffset := 0;
SieveNum :=0;
CopyPreSieveInSieve;
//normal sieving of first block sieve
i := 1; // start with 3
repeat
while Sieve[i] = 0 do
inc(i);
pr := 2*i+1;
inc(i);
j := ((pr*pr)-1) DIV 2;
if j > High(Sieve) then
BREAK;
repeat
Sieve[j] := 0;
inc(j,pr);
until j > High(Sieve);
until false;
CollectPrimes;
PrimPos := InsertSievePrimes(0);
//correct for SieveNum = 0
CalcSievePrimOfs(PrimPos);
Init0Sieve;
sieveOneBlock;
//now start collect with SieveNum = 1
IF PrimPos < High(sievePrimes) then
repeat
sieveOneBlock;
CollectPrimes;
dec(SieveNum);
PrimPos := InsertSievePrimes(PrimPos);
inc(SieveNum);
until PrimPos > High(sievePrimes);
Init0Sieve;
end;
procedure Init0Sieve;
begin
FoundPrimesTotal :=0;
preSieveOffset := 0;
SieveNum :=0;
CalcSievePrimOfs(High(sievePrimes));
end;
procedure CopyPreSieveInSieve;
var
lmt : NativeInt;
Begin
lmt := preSieveOffset+cSieveSize;
lmt := lmt-(High(preSieve)+1);
IF lmt<= 0 then
begin
Move(preSieve[preSieveOffset],Sieve[0],cSieveSize);
if lmt <> 0 then
inc(preSieveOffset,cSieveSize)
else
preSieveOffset := 0;
end
else
begin
Move(preSieve[preSieveOffset],Sieve[0],cSieveSize-lmt);
Move(preSieve[0],Sieve[cSieveSize-lmt],lmt);
preSieveOffset := lmt
end;
end;
procedure sieveOneSieve;
var
sp:tpSievePrim;
pSieve :pByte;
i,j,pr,sn,dSievNum :NativeUint;
Begin
pr := 0;
sn := sieveNum;
sp := @sievePrimes[0];
pSieve := @Sieve[0];
For i := SieveMaxIdx downto 0 do
with sp^ do
begin
pr := pr+svdeltaPrime;
IF svSivNum = sn then
Begin
j := svSivOfs;
repeat
pSieve[j] := 0;
inc(j,pr);
until j > High(Sieve);
dSievNum := j DIV cSieveSize;
svSivOfs := j-dSievNum*cSieveSize;
svSivNum := sn+dSievNum;
// svSivNum := svSivNum+dSievNum;
end;
inc(sp);
end;
i := SieveMaxIdx+1;
repeat
if i > High(SievePrimes) then
BREAK;
with sp^ do
begin
if svSivNum > sn then
Begin
SieveMaxIdx := I-1;
Break;
end;
pr := pr+svdeltaPrime;
j := svSivOfs;
repeat
Sieve[j] := 0;
inc(j,pr);
until j > High(Sieve);
dSievNum := j DIV cSieveSize;
svSivOfs := j-dSievNum*cSieveSize;
svSivNum := sn+dSievNum;
end;
inc(i);
inc(sp);
until false;
end;
procedure CollectPrimes;
//extract primes to FoundPrimes
var
pSieve : pbyte;
pFound : pWord;
i,idx : NativeUint;
Begin
FoundPrimesOffset := SieveNum*(2*cSieveSize);
FoundPrimesIdx := 0;
pFound :=@FoundPrimes[0];
i := 0;
idx := 0;
IF SieveNum = 0 then
//include small primes used to pre-sieve
Begin
repeat
pFound[idx]:= (smlPrimes[idx]-1) DIV 2;
inc(idx);
until smlPrimes[idx]>maxPreSievePrime;
i := (smlPrimes[idx] -1) DIV 2;
end;
//grabbing the primes without if then -> reduces time extremly
//primes are born to let branch-prediction fail.
pSieve:= @Sieve[Low(Sieve)];
repeat
//store every value until a prime aka 1 is found
pFound[idx]:= i;
inc(idx,pSieve[i]);
inc(i);
until i>High(Sieve);
FoundPrimesCnt:= idx;
inc(FoundPrimesTotal,Idx);
end;
procedure SieveOneBlock;inline;
begin
CopyPreSieveInSieve;
sieveOneSieve;
CollectPrimes;
inc(SieveNum);
end;
procedure NextSieve;inline;
Begin
SieveOneBlock;
end;
function Nextprime:Uint64;
Begin
result := FoundPrimes[FoundPrimesIdx]*2+1+FoundPrimesOffset;
if (FoundPrimesIdx=0) AND (sievenum = 1) then
inc(result);
inc(FoundPrimesIdx);
If FoundPrimesIdx>= FoundPrimesCnt then
SieveOneBlock;
end;
function PosOfPrime: Uint64;inline;
Begin
result := FoundPrimesTotal-FoundPrimesCnt+FoundPrimesIdx;
end;
function StartCount : Uint64 ;inline;
begin
result := FoundPrimesTotal-FoundPrimesCnt;
end;
function TotalCount :Uint64;inline;
begin
result := FoundPrimesTotal;
end;
function SieveSize :LongInt;inline;
Begin
result := 2*cSieveSize;
end;
function SieveStart:Uint64;inline;
Begin
result := (SieveNum-1)*2*cSieveSize;
end;
procedure InitPrime;inline;
Begin
Init0Sieve;
SieveOneBlock;
end;
begin
preSieveInit;
sievePrimesInit;
InitPrime;
end.
```

- the test program

```
program test;
{$IFDEF FPC}
{$MODE objFPC}{$Optimization ON,ALL}
{$IFEND}
uses
primsieve;
var
cnt,p,lmt : Uint64;
Begin
lmt := 1000*1000*1000;
p := 0;
while TotalCount < lmt do
Begin
NextSieve;
inc(p);
If p AND (4096-1) = 0 then
write(p:8,TotalCount:15,#13);
end;
cnt := StartCount;
repeat
p := NextPrime;
inc(cnt);
until cnt >= lmt;
writeln(cnt:14,p:14);
end.
{
10^n primecount
# 1 4
# 2 25
# 3 168
# 4 1229
# 5 9592
# 6 78498
# 7 664579
# 8 5761455
# 9 50847534
# 10 455052511
# 11 4118054813
# 12 37607912018
}
```

- @home:

//4.4 Ghz Ryzen 5600G fpc 3.3.1 -O3 -Xs 50847534 999999937 real 0m0,386s 455052511 9999999967 real 0m4,702s 4118054813 99999999977 real 1m11,085s 37607912018 999999999989 real 19m1,195s

### alternative

The main intention is the use in http://rosettacode.org/wiki/Emirp_primes. The speed is about 3x times slower than sieve of Atkin.About 13 secs for 10 billion/146 secs for 100 billion in 64-Bit. But i can hold all primes til 1e11 in 2.5 Gb memory.Test for isEmirp inserted. 32-bit is slow doing 64-Bit math.Using a dynamic array is slow too in NextPrime.

```
program emirp;
{$IFDEF FPC}
{$MODE DELPHI}
{$OPTIMIZATION ON,REGVAR,PEEPHOLE,CSE,ASMCSE}
{$CODEALIGN proc=8}
// {$R+,V+,O+}
{$ELSE}
{$APPLICATION CONSOLE}
{$ENDIF}
uses
sysutils;
type
tSievenum = NativeUint;
const
cBitSize = SizeOf(tSievenum)*8;
cAndMask = cBitSize-1;
InitPrim :array [0..9] of byte = (2,3,5,7,11,13,17,19,23,29);
(*
{MAXANZAHL = 2*3*5*7*11*13*17*19;*PRIM}
MAXANZAHL :array [0..8] of Longint =(2,6,30,210,2310,30030,
510510,9699690,223092870);
{WIFEMAXLAENGE = 1*2*4*6*10*12*16*18; *(PRIM-1)}
WIFEMAXLAENGE :array [0..8] of longint =(1,2,8,48,480,5760,
92160,1658880,36495360);
*)
//Don't sieve with primes that are multiples of 2..InitPrim[BIS]
BIS = 5;
MaxMulFac = 22; {array [0..9] of byte= (2,4,6,10,14,22,26,34,40,50);}
cMaxZahl = 30030;
cRepFldLen = 5760;
MaxUpperLimit = 100*1000*1000*1000-1;
MAXIMUM = ((MaxUpperLimit-1) DIV cMaxZahl+1)*cMaxZahl;
MAXSUCHE = (((MAXIMUM-1) div cMaxZahl+1)*cRepFldLen-1)
DIV cBitSize;
type
tRpFldIdx = 0..cRepFldLen-1;
pNativeUint = ^ NativeUint;
(* numberField as Bit array *)
tsearchFld = array of tSievenum;
tSegment = record
dOfs,
dSegment :tSievenum;
end;
tpSegment = ^tSegment;
tMulFeld = array [0..MaxMulFac shr 1 -1] of tSegment;
tnumberField= array [0..cMaxZahl-1] of word; //word-> 0..cRepFldLen-1
tRevIdx = array [tRpFldIdx] of word;//word-> 0..cMaxZahl-1
tDiffFeld = array [tRpFldIdx] of byte;
tNewPosFeld = array [tRpFldIdx] of Uint64;
tRecPrime = record
rpPrime,
rpsvPos : Uint64;
rpOfs,
rpSeg :LongWord;
end;
var
BitSet,
BitClr : Array [0..cAndMask] Of NativeUint;
deltaNewPos : tNewPosFeld;
MulFeld : tMulFeld;
searchFld : tsearchFld;
number : tnumberField;
DiffFld : tDiffFeld;
RevIdx : tRevIdx;
actSquare : Uint64;
NewStartPos,
MaxPos : Uint64;
const
//K1 = $0101010101010101;
K55 = $5555555555555555;
K33 = $3333333333333333;
KF1 = $0F0F0F0F0F0F0F0F;
KF2 = $00FF00FF00FF00FF;
KF4 = $0000FFFF0000FFFF;
KF8 = $00000000FFFFFFFF;
function popcnt(n:Uint64):integer;overload;inline;
var
c,b,k : NativeUint;
begin
b := n;
k := NativeUint(K55);c := (b shr 1) AND k; b := (b AND k)+C;
k := NativeUint(K33);c := ((b shr 2) AND k);b := (b AND k)+C;
k := NativeUint(KF1);c := ((b shr 4) AND k);b := (b AND k)+c;
k := NativeUint(KF2);c := ((b shr 8) AND k);b := (b AND k)+c;
k := NativeUint(KF4);c := ((b shr 16) AND k);b := (b AND k)+c;
k := NativeUint(KF8);c := (b shr 32)+(b AND k);
result := c;
end;
function popcnt(n:LongWord):integer;overload;
var
c,k : LongWord;
begin
result := n;
IF result = 0 then
EXIT;
k := LongWord(K55);c := (result shr 1) AND k; result := (result AND k)+C;
k := LongWord(K33);c := ((result shr 2) AND k);result := (result AND k)+C;
k := LongWord(KF1);c := ((result shr 4) AND k);result := (result AND k)+c;
k := LongWord(KF2);c := ((result shr 8) AND k);result := (result AND k)+c;
k := LongWord(KF4);
result := (result shr 16) AND k +(result AND k);
end;
procedure Init;
{simple sieve of erathosthenes only eliminating small primes}
var
pr,i,j,Ofs : NativeUint;
Begin
//Init Bitmasks
j := 1;
For i := 0 to cAndMask do
Begin
BitSet[i] := J;
BitClr[i] := NativeUint(NOT(J));
j:= j+j;
end;
//building number wheel excluding multiples of small primes
Fillchar(number,SizeOf(number),#0);
For i := 0 to BIS do
Begin
pr := InitPrim[i];
j := (High(number) div pr)*pr;
repeat
number[j] := 1;
dec(j,pr);
until j <= 0;
end;
// build reverse Index and save distances
i := 1;
j := 0;
RevIdx[0]:= 1;
repeat
Ofs :=0;
repeat
inc(i);
inc(ofs);
until number[i] = 0;
DiffFld[j] := ofs;
inc(j);
RevIdx[j] := i;
until i = High(number);
DiffFld[j] := 2;
//calculate a bitnumber-index into cRepFldLen
Fillchar(number,SizeOf(number),#0);
Ofs := 1;
for i := 0 to cRepFldLen-2 do
begin
inc(Ofs,DiffFld[i]);
number[ofs] := i+1;
end;
//direct index into Mulfeld 2->0 ,4-> 1 ...
For i := 0 to cRepFldLen-1 do
Begin
j := (DiffFld[i] shr 1) -1;
DiffFld[i] := j;
end;
end;
function CalcPos(m: Uint64): Uint64;
{search right position of m}
var
i,res : NativeUint;
Begin
res := m div cMaxZahl;
i := m-res* Uint64(cMaxzahl);//m mod cMaxZahl
while (number[i]= 0) and (i <>1) do
begin
iF i = 0 THEN
begin
Dec(res,cRepFldLen);
i := cMaxzahl;
end;
dec(i);
end; {while}
CalcPos := res *Uint64(cRepFldLen) +number[i];
end;
procedure CalcSqrOfs(out Segment,Ofs :Uint64);
Begin
Segment := actSquare div cMaxZahl;
Ofs := actSquare-Segment*cMaxZahl; //ofs Mod cMaxZahl
Segment := Segment*cRepFldLen;
end;
procedure MulTab(sievePr:Nativeint);
var
k,Segment,Segment0,Rest,Rest0: NativeUint;
Begin
{multiplication-table of differences}
{2* sievePr,4* ,6* ...MaxMulFac*sievePr }
sievePr := sievePr+sievePr;
Segment0 := sievePr div cMaxzahl;
Rest0 := sievePr-Segment0*cMaxzahl;
Segment0 := Segment0 * cRepFldLen;
Segment := Segment0;
Rest := Rest0;
with MulFeld[0] do
begin
dOfs := Rest0;
dSegment:= Segment0;
end;
for k := 1 to MaxMulFac shr 1-1 do
begin
Segment := Segment+Segment0;
Rest := Rest+Rest0;
IF Rest >= cMaxzahl then
Begin
Rest:= Rest-cMaxzahl;
Segment := Segment+cRepFldLen;
end;
with MulFeld[k] do
begin
dOfs := Rest;
dSegment:= Segment;
end;
end;
end;
procedure CalcDeltaNewPos(sievePr,MulPos:NativeUint);
var
Ofs,Segment,prevPos,actPos : Uint64;
i: NativeInt;
Begin
MulTab(sievePr);
//start at sqr sievePrime
CalcSqrOfs(Segment,Ofs);
NewStartPos := Segment+number[Ofs];
prevPos := NewStartPos;
deltaNewPos[0]:= prevPos;
For i := 0 to cRepFldLen-2 do
begin
inc(mulpos);
IF mulpos >= cRepFldLen then
mulpos := 0;
With MulFeld[DiffFld[mulpos]] do
begin
Ofs:= Ofs+dOfs;
Segment := Segment+dSegment;
end;
If Ofs >= cMaxZahl then
begin
Ofs := Ofs-cMaxZahl;
Segment := Segment+cRepFldLen;
end;
actPos := Segment+number[Ofs];
deltaNewPos[i]:= actPos - prevPos;
IF actPos> maxPos then
BREAK;
prevPos := actPos;
end;
deltaNewPos[cRepFldLen-1] := NewStartPos+cRepFldLen*sievePr-prevPos;
end;
procedure SieveByOnePrime(var sf:tsearchFld;sievePr:NativeUint);
var
pNewPos : ^Uint64;
pSiev0,
pSiev : ^tSievenum;// dynamic arrays are slow
Ofs : Int64;
Position : UINt64;
i: NativeInt;
Begin
pSiev0 := @sf[0];
Ofs := MaxPos-sievePr *cRepFldLen;
Position := NewStartPos;
{unmark multiples of sieve prime}
repeat
IF Position < Ofs then
Begin
pNewPos:= @deltaNewPos[0];
For i := Low(deltaNewPos) to High(deltaNewPos) do
Begin
pSiev := pSiev0;
inc(pSiev,Position DIV cBitSize);
//pSiev^ == @sf[Position DIV cBitSize]
pSiev^ := pSiev^ AND BitCLR[Position AND cAndMask];
inc(Position,pNewPos^);
inc(pNewPos);
end
end
else
Begin
pNewPos:= @deltaNewPos[0];
For i := Low(deltaNewPos) to High(deltaNewPos) do
Begin
IF Position >= MaxPos then
Break;
pSiev := pSiev0;
inc(pSiev,Position DIV cBitSize);
pSiev^ := pSiev^ AND BitCLR[Position AND cAndMask];
inc(Position,pNewPos^);
inc(pNewPos);
end
end;
until Position >= MaxPos;
end;
procedure SieveAll;
var
i,
sievePr,
PrimPos,
srPrPos : NativeUint;
Begin
Init;
MaxPos := CalcPos(MaxUpperLimit);
{start of prime sieving}
i := (MaxPos-1) DIV cBitSize+1;
setlength(searchFld,i);
IF Length(searchFld) <> i then
Begin
writeln('Not enough memory');
Halt(-227);
end;
For i := High(searchFld) downto 0 do
searchFld[i] := NativeUint(-1);
{the first prime}
srPrPos := 0;
PrimPos := 0;
sievePr := 1;
actSquare := sievePr;
repeat
{next prime}
inc(srPrPos);
i := 2*(DiffFld[PrimPos]+1);
//binom (a+b)^2; a^2 already known
actSquare := actSquare+(2*sievePr+i)*i;
inc(sievePr,i);
IF actSquare > MaxUpperLimit THEN
BREAK;
{if sievePr == prime then sieve with sievePr}
if BitSet[srPrPos AND cAndMask] AND
searchFld[srPrPos DIV cBitSize] <> 0then
Begin
write(sievePr:8,#8#8#8#8#8#8#8#8);
CalcDeltaNewPos(sievePr,PrimPos);
SieveByOnePrime(searchFld,sievePr);
end;
inc(PrimPos);
if PrimPos = cRepFldLen then
dec(PrimPos,PrimPos);// := 0;
until false;
end;
function InitRecPrime(pr: UInt64):tRecPrime;
var
svPos,sg : NativeUint;
Begin
svPos := CalcPos(pr);
sg := svPos DIV cRepFldLen;
with result do
Begin
rpsvPos := svPos;
rpSeg := sg;
rpOfs := svPos - sg*cRepFldLen;
rpPrime := RevIdx[rpOfs]+ sg*cMaxZahl;
end;
end;
function InitPrimeSvPos(svPos: Uint64):tRecPrime;
var
sg : LongWord;
Begin
sg := svPos DIV cRepFldLen;
with result do
Begin
rpsvPos := svPos;
rpSeg := sg;
rpOfs := svPos - sg*cRepFldLen;
rpPrime := RevIdx[rpOfs]+ sg*cMaxZahl;
end;
end;
function NextPrime(var pr: tRecPrime):Boolean;
var
ofs : LongWord;
svPos : Uint64;
Begin
with pr do
Begin
svPos := rpsvPos;
Ofs := rpOfs;
repeat
inc(svPos);
if svPos > MaxPos then
Begin
result := false;
EXIT;
end;
inc(Ofs);
IF Ofs >= cRepFldLen then
Begin
ofs := 0;
inc(rpSeg);
end;
until BitSet[svPos AND cAndMask] AND
searchFld[svPos DIV cBitSize] <> 0;
rpPrime := rpSeg*Uint64(cMaxZahl)+RevIdx[Ofs];
rpSvPos := svPos;
rpOfs := Ofs;
end;
result := true;
end;
function GetNthPrime(n: Uint64):tRecPrime;
var
i : longWord;
cnt: Uint64;
Begin
IF n > MaxPos then
EXIT;
i := 0;
cnt := Bis;
For i := 0 to n DIV cBitSize do
inc(cnt,PopCnt(NativeUint(searchFld[i])));
i := n DIV cBitSize+1;
while cnt < n do
Begin
inc(cnt,PopCnt(NativeUint(searchFld[i])));
inc(i);
end;
dec(i);
dec(cnt,PopCnt(NativeUint(searchFld[i])));
result := InitPrimeSvPos(i*Uint64(cBitSize)-1);
while cnt < n do
IF NextPrime(Result) then
inc(cnt)
else
Break;
end;
procedure ShowPrimes(loLmt,HiLmt: NativeInt);
var
p1 :tRecPrime;
Begin
IF HiLmt < loLmt then
exit;
p1 := InitRecPrime(loLmt);
while p1.rpPrime < LoLmt do
IF Not(NextPrime(p1)) Then
EXIT;
repeat
write(p1.rpPrime,' ');
IF Not(NextPrime(p1)) Then
Break;
until p1.rpPrime > HiLmt;
writeln;
end;
function CountPrimes(loLmt,HiLmt: NativeInt):LongWord;
var
p1 :tRecPrime;
Begin
result := 0;
IF HiLmt < loLmt then
exit;
p1 := InitRecPrime(loLmt);
while p1.rpPrime < LoLmt do
IF Not(NextPrime(p1)) Then
EXIT;
repeat
inc(result);
IF Not(NextPrime(p1)) Then
Break;
until p1.rpPrime > HiLmt;
end;
procedure WriteCntSmallPrimes(n: NativeInt);
var
i, p,prPos,svPos : nativeUint;
Begin
dec(n);
IF n < 0 then
EXIT;
write('First ',n+1,' primes ');
IF n < Bis then
Begin
For i := 0 to n do
write(InitPrim[i]:3);
end
else
Begin
For i := 0 to BIS do
write(InitPrim[i],' ');
dec(n,Bis);
svPos := 0;
PrPos := 0;
p := 1;
while n> 0 do
Begin
{next prime}
inc(svPos);
inc(p,2*(DiffFld[prPos]+1));
if BitSet[svPos AND cAndMask] AND searchFld[svPos DIV cBitSize] <>0 then
Begin
write(p,' ');
dec(n);
end;
inc(prPos);
if prPos = cRepFldLen then
dec(prPos,prPos);// := 0;
end;
end;
writeln;
end;
function RvsNumL(var n: Uint64):Uint64;
//reverse and last digit, most of the time n > base therefor repeat
const
base = 10;
var
q, c: Int64;
Begin
result := n;
q := 0;
repeat
c:= result div Base;
q := result+ (q-c)*Base;
result := c;
until result < Base;
n := q*Base+result;
end;
function IsEmirp(n:Uint64):boolean;
var
lastDgt:NativeUint;
ofs: NativeUint;
seg : Uint64;
Begin
seg := n;
lastDgt:= RvsNumL(n);
result:= false;
IF (seg = n) OR (n> MaxUpperLimit) then
EXIT;
IF lastDgt in [1,3,7,9] then
Begin
seg := n div cMaxZahl;
ofs := n-seg* cMaxzahl;//m mod cMaxZahl
IF (Number[ofs] <> 0) OR (ofs=1) then
begin
seg := seg *cRepFldLen+number[ofs];
result := BitSet[seg AND cAndMask] AND searchFld[seg DIV cBitSize] <> 0;
end
end;
end;
function GetEmirps(loLmt,HiLmt: Uint64):NativeInt;
var
p1 :tRecPrime;
Begin
result := 0;
IF HiLmt < loLmt then
exit;
IF loLmt > MaxUpperLimit then
Exit;
IF HiLmt > MaxUpperLimit then
HiLmt := MaxUpperLimit;
p1 := InitRecPrime(loLmt);
while p1.rpPrime < LoLmt do
IF Not(NextPrime(p1)) Then
EXIT;
repeat
if isEmirp(p1.rpPrime) then
inc(result);
iF not(NextPrime(p1)) then
BREAK;
until p1.rpPrime > HiLmt;
end;
var
T1,T0: TDateTime;
Anzahl :Uint64;
i,j,dgtCnt,totalCnt : Uint64;
n : LongInt;
Begin
T0 := now;
SieveAll;
T1 := now;
writeln(' ');
Writeln('time for sieving ',FormatDateTime('NN:SS.ZZZ',T1-T0));
Anzahl := BIS;
For n := MaxPos DIV cBitSize-1 downto 0 do
inc(Anzahl,PopCnt(NativeUint(searchFld[n])));
n := MaxPos AND cAndMask;
IF n >0 then
Begin
dec(n);
repeat
IF BitSet[n] AND searchFld[MaxPos DIV cBitSize] <> 0 then
inc(Anzahl);
dec(n);
until n< 0;
end;
Writeln('there are ',Anzahl,' primes til ',MaxUpperLimit);
WriteCntSmallPrimes(20);
write('primes between 100 and 150: ');
ShowPrimes(100,150);
write('count of primes between 7700 and 8000 ');
Writeln(CountPrimes(7700,8000));
i := 100;
repeat
Writeln('the ',i, ' th prime ',GetNthPrime(i).rpPrime);
i := i * 10;
until i*25 > MaxUpperLimit;
writeln;
writeln('Count Emirps');
writeln(' Emirp Total');
writeln('Decimals Count Count');
totalCnt := 0;
j := 10;
i := 2;
dgtCnt := 2; // 13 is not present so 13<->31 isnt found
repeat
write(i:8);
inc(dgtCnt,GetEmirps( j, j+j-1));//10..00->19..99
inc(dgtCnt,GetEmirps(3*j,3*j+j-1));//30..00->39..99
inc(dgtCnt,GetEmirps(7*j,7*j+j-1));//70..00->79..99
inc(dgtCnt,GetEmirps(9*j,9*j+j-1));//90..00->99..99
inc(TotalCnt,dgtCnt);
writeln(dgtCnt:12,TotalCnt:14);
j:=j*10;
inc(i);
dgtCnt := 0;
until j >= MaxUpperLimit;
end.
```

- output

//64-Bit time ./emirp time for sieving 04:17.895 there are 4118054813 primes til 99999999999 First 20 primes 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 primes between 100 and 150: 101 103 107 109 113 127 131 137 139 149 count of primes between 7700 and 8000 30 the 100 th prime 541 the 1000 th prime 7919 the 10000 th prime 104729 the 100000 th prime 1299709 the 1000000 th prime 15485863 the 10000000 th prime 179424673 the 100000000 th prime 2038074743 the 1000000000 th prime 22801763489 Count Emirps Emirp Total Decimals Count Count 2 8 8 3 28 36 4 204 240 5 1406 1646 6 9538 11184 7 70474 81658 8 535578 617236 9 4192024 4809260 10 33619380 38428640 11 274890232 313318872 real 7m44.649s

## Perl

Two examples of pure Perl extensible generators are shown in the Sieve of Eratosthenes#Extensible_sieves section.

The Math::Prime::Util module provides a highly performant, feature-rich library for generating, testing, and manipulating prime numbers in Perl. It offers full interoperability with Perl's bigint pragma.

Limits with a 64-bit Perl:

- nth_prime takes about 20 seconds to return the 10^14th prime and should be fast for all results up to ~4e17. It will be impractically slow past that.
- prime_count uses the LMO algorithm and takes about 35 seconds to return the count for primes to 10^16, and should have state of the art speed to 2^64-1. After that it will use a primality test in the interval so it still useful for large sizes with a small range.
- fast approximations and upper/lower limits are available, which should be fast for any input size including bigints.
`primes`,`next_prime`,`prev_prime`,`forprimes`,`prime_iterator`,`prime_iterator_object`, and primality tests will work for practically any size input. The Math::Prime::Uti::GMP module is recommended for large inputs. With that module, these functions will work quickly for multi-thousand digit numbers.

```
use Math::Prime::Util qw(nth_prime prime_count primes);
# Direct solutions.
# primes([start],end) returns an array reference with all primes in the range
# prime_count([start],end) uses sieving or LMO to return fast prime counts
# nth_prime(n) does just that. It runs quite fast for native size inputs.
say "First 20: ", join(" ", @{primes(nth_prime(20))});
say "Between 100 and 150: ", join(" ", @{primes(100,150)});
say prime_count(7700,8000), " primes between 7700 and 8000";
say "${_}th prime: ", nth_prime($_) for map { 10**
```