Cubic special primes

From Rosetta Code
Cubic special primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Task

n   is smallest prime such that the difference of successive terms are the smallest cubics of positive integers, where     n   <   15000.

11l[edit]

Translation of: FreeBASIC
F is_prime(a)
   I a == 2
      R 1B
   I a < 2 | a % 2 == 0
      R 0B
   L(i) (3 .. Int(sqrt(a))).step(2)
      I a % i == 0
         R 0B
   R 1B

V p = 2
V n = 1
print(2, end' ‘ ’)

L p + n ^ 3 < 15000
   I is_prime(p + n ^ 3)
      p += n ^ 3
      n = 1
      print(p, end' ‘ ’)
   E
      n++
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419 

Action![edit]

INCLUDE "H6:SIEVE.ACT"

PROC Main()
  DEFINE MAX="14999"
  BYTE ARRAY primes(MAX+1)
  INT i=[2],next,count=[1],n=[1]

  Put(125) PutE() ;clear the screen
  Sieve(primes,MAX+1)
  PrintI(i)
  DO
    next=i+n*n*n
    IF next>=MAX THEN
      EXIT
    FI
    IF primes(next) THEN
      n=1 i=next count==+1
      Put(32) PrintI(i)
    ELSE
      n==+1
    FI
  OD
  PrintF("%E%EThere are %I cubic special primes",count)  
RETURN
Output:

Screenshot from Atari 8-bit computer

2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

There are 23 cubic special primes

ALGOL 68[edit]

BEGIN # find a sequence of primes where the members differ by a cube #
    INT max prime = 15 000;
    # sieve the primes to max prime #
    PR read "primes.incl.a68" PR
    []BOOL prime = PRIMESIEVE max prime;
    # construct a table of cubes, we will need at most the cube root of max prime #
    [ 1 : ENTIER exp( ln( max prime ) / 3 ) ]INT cube;
    FOR i TO UPB cube DO cube[ i ] := i * i * i OD;
    # find the prime sequence #
    print( ( "2" ) );
    INT p := 2;
    WHILE p < max prime DO
        # find a prime that is p + a cube #
        INT q := 0;
        FOR i WHILE q := p + cube[ i ];
                    IF q > max prime THEN FALSE ELSE NOT prime[ q ] FI
        DO SKIP OD;
        IF q <= max prime THEN print( ( " ", whole( q, 0 ) ) ) FI;
        p := q
    OD;
    print( ( newline ) )
END
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

ALGOL W[edit]

begin % find a sequence of primes where the members differ by a cube %
    integer MAX_PRIME, MAX_CUBE;
    MAX_PRIME := 15000;
    MAX_CUBE  := entier( exp( ln( MAX_PRIME ) / 3 ) );
    begin
        logical array prime ( 1 :: MAX_PRIME );
        integer array cube  ( 1 :: MAX_CUBE  );
        integer p, q, c;
        % sieve the primes to MAX_PRIME %
        prime( 1 ) := false; prime( 2 ) := true;
        for i := 3 step 2 until MAX_PRIME do prime( i ) := true;
        for i := 4 step 2 until MAX_PRIME do prime( i ) := false;
        for i := 3 step 2 until entier( sqrt( MAX_PRIME ) ) do begin
            integer ii; ii := i + i;
            if prime( i ) then for s := i * i step ii until MAX_PRIME do prime( s ) := false
        end for_i;
        % construct a table of cubes, we will need at most the cube root of max prime %
        for i := 1 until MAX_CUBE do cube( i ) := i * i * i;
        % find the prime sequence %
        writeon( "2" );
        p := 2;
        while p < MAX_PRIME do begin
            % find a prime that is p + a cube %
            c := 1;
            q := p + cube( c );
            while if q > MAX_PRIME then false else not prime( q ) do begin
                c := c + 1;
                q := p + cube( c )
            end while_not_prime_q ;
            if q <= MAX_PRIME then writeon( i_w := 1, s_w := 0, " ", q );
            p := q
        end while_p_lt_MAX_PRIME
    end;
    write()
end.
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

AWK[edit]

# syntax: GAWK -f CUBIC_SPECIAL_PRIMES.AWK
# converted from FreeBASIC
BEGIN {
    start = p = 2
    stop = 15000
    n = 1
    printf("%5d ",p)
    count = 1
    do {
      if (is_prime(p + n^3)) {
        p += n^3
        n = 1
        printf("%5d%1s",p,++count%10?"":"\n")
      }
      else {
        n++
      }
    } while (p + n^3 <= stop)
    printf("\nCubic special primes %d-%d: %d\n",start,stop,count)
    exit(0)
}
function is_prime(x,  i) {
    if (x <= 1) {
      return(0)
    }
    for (i=2; i<=int(sqrt(x)); i++) {
      if (x % i == 0) {
        return(0)
      }
    }
    return(1)
}
Output:
    2     3    11    19    83  1811  2027  2243  2251  2467
 2531  2539  3539  3547  4547  5059 10891 12619 13619 13627
13691 13907 14419
Cubic special primes 2-15000: 23

F#[edit]

// Cubic Special Primes: Nigel Galloway. March 30th., 2021
let fN=let N=[for n in [0..25]->n*n*n] in let mutable n=2 in (fun g->match List.contains(g-n)N with true->n<-g; true |_->false)
primes32()|>Seq.takeWhile((>)16000)|>Seq.filter fN|>Seq.iter(printf "%d "); printfn ""
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

Factor[edit]

Works with: Factor version 0.98
USING: fry io kernel lists lists.lazy math math.functions
math.primes prettyprint ;

2 [ 1 lfrom swap '[ 3 ^ _ + ] lmap-lazy [ prime? ] lfilter car ]
lfrom-by [ 15000 < ] lwhile [ pprint bl ] leach nl
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

FreeBASIC[edit]

#include once "isprime.bas"

dim as integer p = 2, n = 1

print 2;" ";

do
    if isprime(p + n^3) then
        p += n^3
        n = 1
        print p;" ";
    else
        n += 1
    end if
loop until p + n^3 >= 15000
print
end
Output:

2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

Go[edit]

Translation of: Wren
package main

import (
    "fmt"
    "math"
)

func sieve(limit int) []bool {
    limit++
    // True denotes composite, false denotes prime.
    c := make([]bool, limit) // all false by default
    c[0] = true
    c[1] = true
    // no need to bother with even numbers over 2 for this task
    p := 3 // Start from 3.
    for {
        p2 := p * p
        if p2 >= limit {
            break
        }
        for i := p2; i < limit; i += 2 * p {
            c[i] = true
        }
        for {
            p += 2
            if !c[p] {
                break
            }
        }
    }
    return c
}

func isCube(n int) bool {
    s := int(math.Cbrt(float64(n)))
    return s*s*s == n
}

func commas(n int) string {
    s := fmt.Sprintf("%d", n)
    if n < 0 {
        s = s[1:]
    }
    le := len(s)
    for i := le - 3; i >= 1; i -= 3 {
        s = s[0:i] + "," + s[i:]
    }
    if n >= 0 {
        return s
    }
    return "-" + s
}

func main() {
    c := sieve(14999)
    fmt.Println("Cubic special primes under 15,000:")
    fmt.Println(" Prime1  Prime2    Gap  Cbrt")
    lastCubicSpecial := 3
    gap := 1
    count := 1
    fmt.Printf("%7d %7d %6d %4d\n", 2, 3, 1, 1)
    for i := 5; i < 15000; i += 2 {
        if c[i] {
            continue
        }
        gap = i - lastCubicSpecial
        if isCube(gap) {
            cbrt := int(math.Cbrt(float64(gap)))
            fmt.Printf("%7s %7s %6s %4d\n", commas(lastCubicSpecial), commas(i), commas(gap), cbrt)
            lastCubicSpecial = i
            count++
        }
    }
    fmt.Println("\n", count+1, "such primes found.")
}
Output:
Same as Wren example.

jq[edit]

Adapted from Julia

Works with: jq

Works with gojq, the Go implementation of jq

For the definition of `is_prime` used here, see https://rosettacode.org/wiki/Additive_primes
# Output: an array
def cubic_special_primes:
  . as $N
  | sqrt as $maxidx
  | {cprimes: [2], q: 0}
  | until( .cprimes[-1] >= $N or .q >= $N;
      label $out
      | foreach range(1; $maxidx + 1) as $i (.;
          .q = (.cprimes[-1] + ($i * $i * $i))
          | if .q >= $N
	    then .emit = true
            elif .q | is_prime
            then .cprimes = .cprimes + [.q]
            | .emit = true
            else .
	    end;
	    select(.emit)) | {cprimes, q}, break $out )
  | .cprimes ;	 
 
15000
| ("Cubic special primes < \(.):",
    cubic_special_primes)
Output:
[2,3,11,19,83,1811,2027,2243,2251,2467,2531,2539,3539,3547,4547,5059,10891,12619,13619,13627,13691,13907,14419]

Julia[edit]

using Primes

function cubicspecialprimes(N = 15000)
    pmask = primesmask(1, N)
    cprimes, maxidx = [2], isqrt(N)
    while (n = cprimes[end]) < N
        for i in 1:maxidx
            q = n + i * i * i
            if  q > N
                return cprimes
            elseif pmask[q]  # got next qprime
                push!(cprimes, q)
                break
            end
        end
    end
end

println("Cubic special primes < 15000:")
foreach(p -> print(rpad(p[2], 6), p[1] % 10 == 0 ? "\n" : ""), enumerate(cubicspecialprimes()))
Output:
Cubic special primes < 16000:
2     3     11    19    83    1811  2027  2243  2251  2467  
2531  2539  3539  3547  4547  5059  10891 12619 13619 13627 
13691 13907 14419

Mathematica/Wolfram Language[edit]

start = {2};
Do[
 If[PrimeQ[i],
  last = Last[start];
  If[IntegerQ[Surd[i - last, 3]],
   AppendTo[start, i]
   ]
  ]
 ,
 {i, 3, 15000}
 ]
start
Output:
{2,3,11,19,83,1811,2027,2243,2251,2467,2531,2539,3539,3547,4547,5059,10891,12619,13619,13627,13691,13907,14419}

Nim[edit]

Translation of: Go
import math, strformat

func sieve(limit: Positive): seq[bool] =
  # True denotes composite, false denotes prime.
  result = newSeq[bool](limit + 1)   # All false by default.
  result[0] = true
  result[1] = true
  # No need to bother with even numbers over 2 for this task.
  var p = 3
  while true:
    let p2 = p * p
    if p2 > limit: break
    for i in countup(p2, limit, 2 * p):
      result[i] = true
    while true:
      inc p, 2
      if not result[p]: break

func isCube(n: int): bool =
  let s = cbrt(n.toFloat).int
  result = s * s * s == n

let c = sieve(14999)
echo "Cubic special primes under 15_000:"
echo " Prime1  Prime2    Gap  Cbrt"
var lastCubicSpecial = 3
var count = 1
echo &"{2:7} {3:7} {1:6} {1:4}"
for n in countup(5, 14999, 2):
  if c[n]: continue
  let gap = n - lastCubicSpecial
  if gap.isCube:
    let gapCbrt = cbrt(gap.toFloat).int
    echo &"{lastCubicSpecial:7} {n:7} {gap:6} {gapCbrt:4}"
    lastCubicSpecial = n
    inc count
echo &"\n{count + 1} such primes found."
Output:
Cubic special primes under 15_000:
 Prime1  Prime2    Gap  Cbrt
      2       3      1    1
      3      11      8    2
     11      19      8    2
     19      83     64    4
     83    1811   1728   12
   1811    2027    216    6
   2027    2243    216    6
   2243    2251      8    2
   2251    2467    216    6
   2467    2531     64    4
   2531    2539      8    2
   2539    3539   1000   10
   3539    3547      8    2
   3547    4547   1000   10
   4547    5059    512    8
   5059   10891   5832   18
  10891   12619   1728   12
  12619   13619   1000   10
  13619   13627      8    2
  13627   13691     64    4
  13691   13907    216    6
  13907   14419    512    8

23 such primes found.

Perl[edit]

Library: ntheory
use strict;
use warnings;
use feature 'say';
use  ntheory 'is_prime';

my @sp = my $previous = 2;
do {
    my($next,$n);
    while () { last if is_prime( $next = $previous + ++$n**3 ) }
    push @sp, $next;
    $previous = $next;
} until $sp[-1] >= 15000;

pop @sp and say join ' ', @sp;
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

Phix[edit]

See Quadrat_Special_Primes#Phix


Python[edit]

#!/usr/bin/python

def isPrime(n):
    for i in range(2, int(n**0.5) + 1):
        if n % i == 0:
            return False        
    return True

if __name__ == '__main__':
    p = 2
    n = 1

    print("2",end = " ")
    while True:
        if isPrime(p + n**3):
            p += n**3
            n = 1
            print(p,end = " ")
        else:
            n += 1
        if p + n**3 >= 15000:
            break
Output:
2  3  11  19  83  1811  2027  2243  2251  2467  2531  2539  3539  3547  4547  5059  10891  12619  13619  13627  13691  13907  14419


Raku[edit]

A two character difference from the Quadrat Special Primes entry. (And it could have been one.)

my @sqp = 2, -> $previous {
    my $next;
    for (1..∞).map: *³ {
        $next = $previous + $_;
        last if $next.is-prime;
    }
    $next
} … *;

say "{+$_} matching numbers:\n", $_».fmt('%5d').batch(7).join: "\n" given
    @sqp[^(@sqp.first: * > 15000, :k)];
Output:
23 matching numbers:
    2     3    11    19    83  1811  2027
 2243  2251  2467  2531  2539  3539  3547
 4547  5059 10891 12619 13619 13627 13691
13907 14419

REXX[edit]

/*REXX program finds the smallest primes such that the difference of successive terms   */
/*───────────────────────────────────────────────────── are the smallest cubic numbers. */
parse arg hi cols .                              /*obtain optional argument from the CL.*/
if   hi=='' |   hi==","  then   hi= 15000        /* "      "         "   "   "     "    */
if cols=='' | cols==","  then cols=    10        /* "      "         "   "   "     "    */
call genP                                        /*build array of semaphores for primes.*/
w= 10                                            /*width of a number in any column.     */
                    title= 'the smallest primes  < '    commas(hi)     " such that the" ,
                           'difference of successive terma are the smallest cubic numbers'
if cols>0 then say ' index │'center(title,  1 + cols*(w+1)     )    /*display the title.*/
if cols>0 then say '───────┼'center(""   ,  1 + cols*(w+1), '─')    /*   "     "    sep.*/
found= 0;             idx= 1                     /*initialize number of  cbp  and index.*/
op= 1
$=                                               /*a list of  cubic  primes  (so far).  */
     do j=0  by 0
                      do k=1  until !.np
                      np= op + k**3              /*find the next cube plus the oldPrime.*/
                      if np>= hi  then leave j   /*Is newPrime too big?  Then quit.     */
                      end   /*k*/
     found= found + 1                            /*bump number of primes of this type.  */
     op= np                                      /*assign the newPrime  to the  oldPrime*/
     if cols<0           then iterate            /*Build the list  (to be shown later)? */
     c= commas(np)                               /*maybe add commas to the number.      */
     $= $ right(c, max(w, length(c) ) )          /*add a cubic prime──►list, allow big#.*/
     if found//cols\==0  then iterate            /*have we populated a line of output?  */
     say center(idx, 7)'│'  substr($, 2);   $=   /*display what we have so far  (cols). */
     idx= idx + cols                             /*bump the  index  count for the output*/
     end   /*j*/

if $\==''  then say center(idx, 7)"│"  substr($, 2)  /*possible display residual output.*/
if cols>0 then say '───────┴'center(""   ,  1 + cols*(w+1), '─')    /*display foot sep. */
say
say 'Found '       commas(found)         " of "       title
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: !.= 0                                      /*placeholders for primes (semaphores).*/
      @.1=2;  @.2=3;  @.3=5;  @.4=7;  @.5=11     /*define some low primes.              */
      !.2=1;  !.3=1;  !.5=1;  !.7=1;  !.11=1     /*   "     "   "    "     flags.       */
                        #=5;     s.#= @.# **2    /*number of primes so far;     prime². */
                                                 /* [↓]  generate more  primes  ≤  high.*/
        do j=@.#+2  by 2  to hi                  /*find odd primes from here on.        */
        parse var j '' -1 _; if     _==5  then iterate  /*J divisible by 5?  (right dig)*/
                             if j// 3==0  then iterate  /*"     "      " 3?             */
                             if j// 7==0  then iterate  /*"     "      " 7?             */
               do k=5  while s.k<=j              /* [↓]  divide by the known odd primes.*/
               if j // @.k == 0  then iterate j  /*Is  J ÷ X?  Then not prime.     ___  */
               end   /*k*/                       /* [↑]  only process numbers  ≤  √ J   */
        #= #+1;    @.#= j;    s.#= j*j;   !.j= 1 /*bump # of Ps; assign next P;  P²; P# */
        end          /*j*/;   return
output   when using the default inputs:
 index │  the smallest primes  <  15,000  such that the difference of successive terma are the smallest cubic numbers
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1   │          2          3         11         19         83      1,811      2,027      2,243      2,251      2,467
  11   │      2,531      2,539      3,539      3,547      4,547      5,059     10,891     12,619     13,619     13,627
  21   │     13,691     13,907     14,419
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  23  of  the smallest primes  <  15,000  such that the difference of successive terma are the smallest cubic numbers

Ring[edit]

Also see Quadrat_Special_Primes#Ring

load "stdlib.ring"

see "working..." + nl

Primes = []
limit1 = 50
oldPrime = 2
add(Primes,2)

for n = 1 to limit1
    nextPrime = oldPrime + pow(n,3)
    if isprime(nextPrime)
       n = 1
       add(Primes,nextPrime)
       oldPrime = nextPrime
    else
       nextPrime = nextPrime - oldPrime
    ok
next

see "prime1 prime2 Gap Cbrt" + nl
for n = 1 to Len(Primes)-1
    diff = Primes[n+1] - Primes[n]
    for m = 1 to diff
        if pow(m,3) = diff
           cbrt = m
           exit
        ok
    next
    see ""+ Primes[n] + "      " + Primes[n+1] + "    " + diff + "     " + cbrt + nl
next   
 
see "Found " + Len(Primes) + " of the smallest primes < 15,000  such that the difference of successive terma are the smallest cubic numbers" + nl  
 
see "done..." + nl
Output:
working...
prime1 prime2 Gap Cbrt
2      3    1     1
3      11    8     2
11      19    8     2
19      83    64     4
83      1811    1728     12
1811      2027    216     6
2027      2243    216     6
2243      2251    8     2
2251      2467    216     6
2467      2531    64     4
2531      2539    8     2
2539      3539    1000     10
3539      3547    8     2
3547      4547    1000     10
4547      5059    512     8
5059      10891    5832     18
10891      12619    1728     12
12619      13619    1000     10
13619      13627    8     2
13627      13691    64     4
13691      13907    216     6
13907      14419    512     8
Found 23 of the smallest primes < 15,000  such that the difference of successive terma are the smallest cubic numbers
done...

Sidef[edit]

func cubic_primes(callback) {

    var prev = 2
    callback(prev)

    loop {
        var curr = (1..Inf -> lazy.map { prev + _**3 }.first { .is_prime })
        callback(curr)
        prev = curr
    }
}

say gather {
    cubic_primes({|k|
        break if (k >= 15000)
        take(k)
    })
}
Output:
[2, 3, 11, 19, 83, 1811, 2027, 2243, 2251, 2467, 2531, 2539, 3539, 3547, 4547, 5059, 10891, 12619, 13619, 13627, 13691, 13907, 14419]

Wren[edit]

Library: Wren-math
Library: Wren-fmt
import "/math" for Int
import "/fmt" for Fmt

var primes = Int.primeSieve(14999)
System.print("Cubic special primes under 15,000:")
System.print(" Prime1  Prime2    Gap  Cbrt")
var lastCubicSpecial = 3
var gap = 1
var count = 1
Fmt.print("$,7d $,7d $,6d $4d", 2, 3, 1, 1)
for (p in primes.skip(2)) {
    gap = p - lastCubicSpecial
    if (Int.isCube(gap)) {
        Fmt.print("$,7d $,7d $,6d $4d", lastCubicSpecial, p, gap, gap.cbrt.truncate)
        lastCubicSpecial = p
        count = count + 1
    }
}
System.print("\n%(count+1) such primes found.")
Output:
Cubic special primes under 15,000:
 Prime1  Prime2    Gap  Cbrt
      2       3      1    1
      3      11      8    2
     11      19      8    2
     19      83     64    4
     83   1,811  1,728   12
  1,811   2,027    216    6
  2,027   2,243    216    6
  2,243   2,251      8    2
  2,251   2,467    216    6
  2,467   2,531     64    4
  2,531   2,539      8    2
  2,539   3,539  1,000   10
  3,539   3,547      8    2
  3,547   4,547  1,000   10
  4,547   5,059    512    8
  5,059  10,891  5,832   18
 10,891  12,619  1,728   12
 12,619  13,619  1,000   10
 13,619  13,627      8    2
 13,627  13,691     64    4
 13,691  13,907    216    6
 13,907  14,419    512    8

23 such primes found.

XPL0[edit]

func IsPrime(N);        \Return 'true' if N is prime
int  N, I;
[if N <= 2 then return N = 2;
if (N&1) = 0 then \even >2\ return false;
for I:= 3 to sqrt(N) do
    [if rem(N/I) = 0 then return false;
    I:= I+1;
    ];
return true;
];

int N, C, T;
[N:= 2;
repeat  C:= 1;
        loop    [T:= N + C*C*C;
                if IsPrime(T) then
                        [IntOut(0, N);
                        ChOut(0, ^ );
                        N:= T;
                        quit;
                        ]
                else    C:= C+1;
                ];
until   N > 15000;
]
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419