Next special primes

From Rosetta Code
Next 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 is strictly increasing, where     n   <   1050.

AWK

<lang AWK>

  1. syntax: GAWK -f NEXT_SPECIAL_PRIMES.AWK

BEGIN { 

   start = 1
   stop = 1050
   print("Prime1 Prime2 Gap")
   last_special = 3
   last_gap = 1
   printf("%6d %6d %3d\n",2,3,last_gap)
   count = 1
   for (i=start; i<=stop; i++) {
     if (is_prime(i) && i-last_special > last_gap) {
       last_gap = i - last_special
       printf("%6d %6d %3d\n",last_special,i,last_gap)
       last_special = i
       count++
     }
   }
   printf("Next 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)

} </lang>

Output:
Prime1 Prime2 Gap
     2      3   1
     3      5   2
     5     11   6
    11     19   8
    19     29  10
    29     41  12
    41     59  18
    59     79  20
    79    101  22
   101    127  26
   127    157  30
   157    191  34
   191    227  36
   227    269  42
   269    313  44
   313    359  46
   359    409  50
   409    461  52
   461    521  60
   521    587  66
   587    659  72
   659    733  74
   733    809  76
   809    887  78
   887    967  80
   967   1049  82
Next special primes 1-1050: 26

ALGOL W

<lang algolw>begin % find some primes where the gap between the current prtime and the next is greater than % 

     % the gap between previus primes and the current                                         %
   % sets p( 1 :: n ) to a sieve of primes up to n %
   procedure Eratosthenes ( logical array p( * ) ; integer value n ) ;
   begin
       p( 1 ) := false; p( 2 ) := true;
       for i := 3 step 2 until n do p( i ) := true;
       for i := 4 step 2 until n do p( i ) := false;
       for i := 3 step 2 until truncate( sqrt( n ) ) do begin
           integer ii; ii := i + i;
           if p( i ) then for pr := i * i step ii until n do p( pr ) := false
       end for_i ;
   end Eratosthenes ;
   integer MAX_NUMBER;
   MAX_NUMBER := 1050;
   begin
       logical array prime( 1 :: MAX_NUMBER );
       integer       pCount, pGap, thisPrime, nextPrime;
       % sieve the primes to MAX_NUMBER %
       Eratosthenes( prime, MAX_NUMBER );
       % the first gap is 1 (between 2 and 3) the gap between all other primes is even       %
       % so we treat 2-3 as a special case                                                   %
       write( "         this  next" );
       write( "element prime prime  gap" );
       pCount := pGap := 1; thisPrime := 2; nextPrime := 3;
       write( i_w := 5, s_w := 0, "  ", pCount, ":", thisPrime, " ", nextPrime, pGap );
       pGap := 0;
       while thisPrime < MAX_NUMBER do begin
           thisPrime := nextPrime;
           while begin
               pGap      := pGap + 2;
               nextPrime := thisPrime + pGap;
               nextPrime < MAX_NUMBER and not prime( nextPrime )
           end do begin end;
           if nextPrime < MAX_NUMBER then begin
               pCount := pCount + 1;
               write( i_w := 5, s_w := 0, "  ", pCount, ":", thisPrime, " ", nextPrime, pGap )
           end if_nextPrime_lt_MAX_NUMBER
       end while_thisPrime_lt_MAX_NUMBER
   end

end.</lang>

Output:
         this  next
element prime prime  gap
      1:    2     3    1
      2:    3     5    2
      3:    5    11    6
      4:   11    19    8
      5:   19    29   10
      6:   29    41   12
      7:   41    59   18
      8:   59    79   20
      9:   79   101   22
     10:  101   127   26
     11:  127   157   30
     12:  157   191   34
     13:  191   227   36
     14:  227   269   42
     15:  269   313   44
     16:  313   359   46
     17:  359   409   50
     18:  409   461   52
     19:  461   521   60
     20:  521   587   66
     21:  587   659   72
     22:  659   733   74
     23:  733   809   76
     24:  809   887   78
     25:  887   967   80
     26:  967  1049   82

C

<lang c>#include <stdio.h>

  1. include <stdbool.h>

bool isPrime(int n) { 

   int d;
   if (n < 2)  return false;
   if (!(n%2)) return n == 2;
   if (!(n%3)) return n == 3;
   d = 5;
   while (d*d <= n) {
       if (!(n%d)) return false;
       d += 2;
       if (!(n%d)) return false;
       d += 4;
   }
   return true;

}

int main() { 

   int i, lastSpecial = 3, lastGap = 1;
   printf("Special primes under 1,050:\n");
   printf("Prime1 Prime2 Gap\n");
   printf("%6d %6d %3d\n", 2, 3, lastGap);
   for (i = 5; i < 1050; i += 2) {
       if (isPrime(i) && (i-lastSpecial) > lastGap) {
           lastGap = i - lastSpecial;
           printf("%6d %6d %3d\n", lastSpecial, i, lastGap);
           lastSpecial = i;
       }
   }

}</lang>

Output:
Special primes under 1,050:
Prime1 Prime2 Gap
     2      3   1
     3      5   2
     5     11   6
    11     19   8
    19     29  10
    29     41  12
    41     59  18
    59     79  20
    79    101  22
   101    127  26
   127    157  30
   157    191  34
   191    227  36
   227    269  42
   269    313  44
   313    359  46
   359    409  50
   409    461  52
   461    521  60
   521    587  66
   587    659  72
   659    733  74
   733    809  76
   809    887  78
   887    967  80
   967   1049  82

F#

This task uses Extensible Prime Generator (F#) <lang fsharp> // Next special primes. Nigel Galloway: March 26th., 2021 let mP=let mutable n,g=2,0 in primes32()|>Seq.choose(fun y->match y-n>g,n with (true,i)->g<-y-n; n<-y; Some(i,g,y) |_->None) mP|>Seq.takeWhile(fun(_,_,n)->n<1050)|>Seq.iteri(fun i (n,g,l)->printfn "n%d=%d n%d=%d n%d-n%d=%d" i n (i+1) l (i+1) i g) </lang>

Output:
n0=2 n1=3 n1-n0=1
n1=3 n2=5 n2-n1=2
n2=5 n3=11 n3-n2=6
n3=11 n4=19 n4-n3=8
n4=19 n5=29 n5-n4=10
n5=29 n6=41 n6-n5=12
n6=41 n7=59 n7-n6=18
n7=59 n8=79 n8-n7=20
n8=79 n9=101 n9-n8=22
n9=101 n10=127 n10-n9=26
n10=127 n11=157 n11-n10=30
n11=157 n12=191 n12-n11=34
n12=191 n13=227 n13-n12=36
n13=227 n14=269 n14-n13=42
n14=269 n15=313 n15-n14=44
n15=313 n16=359 n16-n15=46
n16=359 n17=409 n17-n16=50
n17=409 n18=461 n18-n17=52
n18=461 n19=521 n19-n18=60
n19=521 n20=587 n20-n19=66
n20=587 n21=659 n21-n20=72
n21=659 n22=733 n22-n21=74
n22=733 n23=809 n23-n22=76
n23=809 n24=887 n24-n23=78
n24=887 n25=967 n25-n24=80
n25=967 n26=1049 n26-n25=82

Go

<lang go>package main

import "fmt"

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 main() { 

   c := sieve(1049)
   fmt.Println("Special primes under 1,050:")
   fmt.Println("Prime1 Prime2 Gap")
   lastSpecial := 3
   lastGap := 1
   fmt.Printf("%6d %6d %3d\n", 2, 3, lastGap)
   for i := 5; i < 1050; i += 2 {
       if !c[i] && (i-lastSpecial) > lastGap {
           lastGap = i - lastSpecial
           fmt.Printf("%6d %6d %3d\n", lastSpecial, i, lastGap)
           lastSpecial = i
       }
   }

}</lang>

Output:
Special primes under 1,050:
Prime1 Prime2 Gap
     2      3   1
     3      5   2
     5     11   6
    11     19   8
    19     29  10
    29     41  12
    41     59  18
    59     79  20
    79    101  22
   101    127  26
   127    157  30
   157    191  34
   191    227  36
   227    269  42
   269    313  44
   313    359  46
   359    409  50
   409    461  52
   461    521  60
   521    587  66
   587    659  72
   659    733  74
   733    809  76
   809    887  78
   887    967  80
   967   1049  82

Java

Translation of: C

<lang java>class SpecialPrimes { 

   private static boolean isPrime(int n) {
       if (n < 2)  return false;
       if (n%2 == 0) return n == 2;
       if (n%3 == 0) return n == 3;
       int d = 5;
       while (d*d <= n) {
           if (n%d == 0) return false;
           d += 2;
           if (n%d == 0) return false;
           d += 4;
       }
       return true;
   }

   public static void main(String[] args) {
       System.out.println("Special primes under 1,050:");
       System.out.println("Prime1 Prime2 Gap");
       int lastSpecial = 3;
       int lastGap = 1;
       System.out.printf("%6d %6d %3d\n", 2, 3, lastGap);
       for (int i = 5; i < 1050; i += 2) {
           if (isPrime(i) && (i-lastSpecial) > lastGap) {
               lastGap = i - lastSpecial;
               System.out.printf("%6d %6d %3d\n", lastSpecial, i, lastGap);
               lastSpecial = i;
           }
       }
   }

}</lang>

Output:
Special primes under 1,050:
Prime1 Prime2 Gap
     2      3   1
     3      5   2
     5     11   6
    11     19   8
    19     29  10
    29     41  12
    41     59  18
    59     79  20
    79    101  22
   101    127  26
   127    157  30
   157    191  34
   191    227  36
   227    269  42
   269    313  44
   313    359  46
   359    409  50
   409    461  52
   461    521  60
   521    587  66
   587    659  72
   659    733  74
   733    809  76
   809    887  78
   887    967  80
   967   1049  82

Julia

<lang julia>using Primes

let 

   println("Special primes under 1050:\nPrime1 Prime2 Gap")
   println("     2     3   1")
   pmask = primesmask(1, 1050)
   n, gap = 3, 2
   while n + gap < 1050
       if pmask[n + gap]
           println(lpad(n, 6), lpad(n + gap, 6), lpad(gap, 4))
           n += gap
       end
       gap += 2
   end

end

</lang>

Output:
Special primes under 1050:
Prime1 Prime2 Gap
     2     3   1
     3     5   2
     5    11   6
    11    19   8
    19    29  10
    29    41  12
    41    59  18
    59    79  20
    79   101  22
   101   127  26
   127   157  30
   157   191  34
   191   227  36
   227   269  42
   269   313  44
   313   359  46
   359   409  50
   409   461  52
   461   521  60
   521   587  66
   587   659  72
   659   733  74
   733   809  76
   809   887  78
   887   967  80
   967  1049  82


Pascal

Library: primTrial

just showing the small difference to increasing prime gaps.
LastPrime is updated outside or inside If

<lang pascal> program NextSpecialprimes; //increasing prime gaps see //https://oeis.org/A002386 https://en.wikipedia.org/wiki/Prime_gap uses 

 sysutils,
 primTrial;

procedure GetIncreasingGaps; var 

 Gap,LastPrime,p : NativeUInt;

Begin 

 InitPrime;
 Writeln('next increasing prime gap');
 writeln('Prime1':8,'Prime2':8,'Gap':4);
 Gap := 0;
 LastPrime := actPrime;
 repeat
   p := NextPrime;
   if p-LastPrime > Gap then
   Begin
     Gap := p-LastPrime;
     writeln(LastPrime:8,P:8,Gap:4);

   end;
   LastPrime := p;
 until LastPrime > 1000;

end;

procedure NextSpecial; var 

 Gap,LastPrime,p : NativeUInt;

Begin 

 InitPrime;
 Writeln('next special prime');
 writeln('Prime1':8,'Prime2':8,'Gap':4);
 Gap := 0;
 LastPrime := actPrime;
 repeat
   p := NextPrime;
   if p-LastPrime > Gap then
   Begin
     Gap := p-LastPrime;
     writeln(LastPrime:8,P:8,Gap:4);
     LastPrime := p;
   end;

 until LastPrime > 1000;

end;

begin 

 GetIncreasingGaps;
 writeln;
 NextSpecial;

end.</lang>

Output:
next increasing prime gap
  Prime1  Prime2 Gap
       2       3   1
       3       5   2
       7      11   4
      23      29   6
      89      97   8
     113     127  14
     523     541  18
     887     907  20

next special prime
  Prime1  Prime2 Gap
       2       3   1
       3       5   2
       5      11   6
      11      19   8
      19      29  10
      29      41  12
      41      59  18
      59      79  20
      79     101  22
     101     127  26
     127     157  30
     157     191  34
     191     227  36
     227     269  42
     269     313  44
     313     359  46
     359     409  50
     409     461  52
     461     521  60
     521     587  66
     587     659  72
     659     733  74
     733     809  76
     809     887  78
     887     967  80
     967    1049  82

Phix

integer lastSpecial = 3, lastGap = 1
printf(1,"Special primes under 1,050:\n")
printf(1,"Prime1 Prime2 Gap\n")
printf(1,"%6d %6d %3d\n", {2, 3, lastGap})
for i=5 to 1050 by 2 do
    if is_prime(i) and (i-lastSpecial) > lastGap then
        lastGap = i - lastSpecial
        printf(1,"%6d %6d %3d\n", {lastSpecial, i, lastGap})
        lastSpecial = i
    end if
end for
Output:
Special primes under 1,050:
Prime1 Prime2 Gap
     2      3   1
     3      5   2
     5     11   6
    11     19   8
    19     29  10
    29     41  12
    41     59  18
    59     79  20
    79    101  22
   101    127  26
   127    157  30
   157    191  34
   191    227  36
   227    269  42
   269    313  44
   313    359  46
   359    409  50
   409    461  52
   461    521  60
   521    587  66
   587    659  72
   659    733  74
   733    809  76
   809    887  78
   887    967  80
   967   1049  82

Raku

<lang perl6>sub is-special ( ($previous, $gap) ) { 

   state @primes = grep *.is-prime, 2..*;
   shift @primes while @primes[0] <= $previous + $gap;
   return ( @primes[0], @primes[0] - $previous );

}

my @specials = (2, 0), &is-special … *;

my $limit = @specials.first: :k, *.[0] > 1050;

say .fmt('%4d') for @specials.head($limit);</lang>

Output:
   2    0
   3    1
   5    2
  11    6
  19    8
  29   10
  41   12
  59   18
  79   20
 101   22
 127   26
 157   30
 191   34
 227   36
 269   42
 313   44
 359   46
 409   50
 461   52
 521   60
 587   66
 659   72
 733   74
 809   76
 887   78
 967   80
1049   82

REXX

Translation of: RING

<lang rexx>/*REXX program finds next special primes: difference of successive terms is increasing.*/ parse arg hi cols . /*obtain optional argument from the CL.*/ if hi== | hi=="," then hi= 1050 /* " " " " " " */ if cols== | cols=="," then cols= 10 /* " " " " " " */ call genP /*build array of semaphores for primes.*/ w= 10 /*width of a number in any column. */ 

                       @nsp= ' next special primes  < '      commas(hi)   ,
                             " such that the different of successive terms is increasing"

if cols>0 then say ' index │'center(@nsp , 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') op= @.1 /*assign oldPrime to the first prime.*/ nsp= 0; idx= 1 /*initialize number of nsp and index.*/ $= /*a list of nice primes (so far). */ 

    do j=0;       np= op + j                    /*assign newPrime to oldPrime  +  j    */
    if np>=hi         then leave                /*Is  newPrimeN ≥ hi?  Then leave loop.*/
    if \!.np          then iterate              /*Is  np  a prime?   Then skip this  J.*/
    nsp= nsp + 1                                /*bump the number of   nsp's.          */
    op= np                                      /*set oldPrime to the value of newPrime*/
    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 nice prime ──► list, allow big#*/
    if nsp//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.*/ say say 'Found ' commas(nsp) @nsp 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?             */
                                                /* [↑]  the above five lines saves time*/
              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</lang>
output   when using the default inputs:
 index │            next special primes  <  1,050  such that the different of successive terms is increasing
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1   │          2          3          5         11         19         29         41         59         79        101
  11   │        127        157        191        227        269        313        359        409        461        521
  21   │        587        659        733        809        887        967      1,049

Found  27  next special primes  <  1,050  such that the different of successive terms is increasing

Ring

<lang ring> load "stdlib.ring"

see "working..." + nl

Primes = [] limit1 = 100 oldPrime = 2

for n = 1 to limit1 

   nextPrime = oldPrime + n
   if isprime(nextPrime)
      add(Primes,nextPrime)
      oldPrime = nextPrime
   ok

next

see "prime1 prime2 Gap" + nl for n = 1 to Len(Primes)-1 

   diff = Primes[n+1] - Primes[n]
   see ""+ Primes[n] + "      " + Primes[n+1] + "    " + diff + nl

next

see nl + "done..." + nl

</lang>

Output:
working...
prime1 prime2 Gap
3      5    2
5      11    6
11      19    8
19      29    10
29      41    12
41      59    18
59      79    20
79      101    22
101      127    26
127      157    30
157      191    34
191      227    36
227      269    42
269      313    44
313      359    46
359      409    50
409      461    52
461      521    60
521      587    66
587      659    72
659      733    74
733      809    76
809      887    78
887      967    80
967      1049    82
done...

Wren

Library: Wren-math
Library: Wren-fmt

<lang ecmascript>import "/math" for Int import "/fmt" for Fmt

var primes = Int.primeSieve(1049) System.print("Special primes under 1,050:") System.print("Prime1 Prime2 Gap") var lastSpecial = primes[1] var lastGap = primes[1] - primes[0] Fmt.print("$6d $6d $3d", primes[0], primes[1], lastGap) for (p in primes.skip(2)) { 

   if ((p - lastSpecial) > lastGap) {
       lastGap = p - lastSpecial
       Fmt.print("$6d $6d $3d", lastSpecial, p, lastGap)
       lastSpecial = p
   }

}</lang>

Output:
Special primes under 1,050:
Prime1 Prime2 Gap
     2      3   1
     3      5   2
     5     11   6
    11     19   8
    19     29  10
    29     41  12
    41     59  18
    59     79  20
    79    101  22
   101    127  26
   127    157  30
   157    191  34
   191    227  36
   227    269  42
   269    313  44
   313    359  46
   359    409  50
   409    461  52
   461    521  60
   521    587  66
   587    659  72
   659    733  74
   733    809  76
   809    887  78
   887    967  80
   967   1049  82
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