I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

# Next special primes

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.

n   is smallest prime such that the difference of successive terms is strictly increasing, where     n   <   1050.

## 11l

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 = 3
V i = 2
print(‘2 3’, end' ‘ ’)

L p + i < 1050
I is_prime(p + i)
p += i
print(p, end' ‘ ’)
i += 2
Output:
2 3 5 11 19 29 41 59 79 101 127 157 191 227 269 313 359 409 461 521 587 659 733 809 887 967 1049

## Action!

INCLUDE "H6:SIEVE.ACT"

PROC Main()
DEFINE MAX="1049"
BYTE ARRAY primes(MAX+1)
INT i,count=[1],lastprime=[3],lastgap=[1]

Put(125) PutE() ;clear the screen
Sieve(primes,MAX+1)
PrintI(lastprime)
FOR i=1 TO MAX
DO
IF primes(i)=1 AND i-lastprime>lastgap THEN
lastgap=i-lastprime
lastprime=i
Put(32) PrintI(i)
count==+1
FI
OD
PrintF("%E%EThere are %I next special primes",count)
RETURN
Output:
3 5 11 19 29 41 59 79 101 127 157 191 227 269 313 359 409 461 521 587 659 733 809 887 967 1049

There are 26 next special primes

## ALGOL W

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.
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

## Arturo

specials: new [2 3]
lim: 1050
lastP: 3
lastGap: 1

loop 5.. .step:2 lim 'n [
if not? prime? n -> continue
if lastGap < n - lastP [
lastGap: n - lastP
lastP: n
'specials ++ n
]
]

print "List of next special primes less than 1050:"
print specials
Output:
List of next special primes less than 1050:
2 3 5 11 19 29 41 59 79 101 127 157 191 227 269 313 359 409 461 521 587 659 733 809 887 967 1049

## AWK

# 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)
}

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

## BASIC

### BASIC256

function isPrime(v)
if v < 2 then return False
if v mod 2 = 0 then return v = 2
if v mod 3 = 0 then return v = 3
d = 5
while d * d <= v
if v mod d = 0 then return False else d += 2
end while
return True
end function

p = 3
i = 2

print "2 3 "; #special case
do
if isPrime(p + i) then
p += i
print p; " ";
end if
i += 2
until p + i >= 1050
end

### PureBasic

Procedure isPrime(v.i)
If v <= 1  : ProcedureReturn #False
ElseIf v < 4  : ProcedureReturn #True
ElseIf v % 2 = 0 : ProcedureReturn #False
ElseIf v < 9  : ProcedureReturn #True
ElseIf v % 3 = 0 : ProcedureReturn #False
Else
Protected r = Round(Sqr(v), #PB_Round_Down)
Protected f = 5
While f <= r
If v % f = 0 Or v % (f + 2) = 0
ProcedureReturn #False
EndIf
f + 6
Wend
EndIf
ProcedureReturn #True
EndProcedure

OpenConsole()
p.i = 3
i.i = 2

Print("2 3 ") ;special Case
Repeat
If isPrime(p + i)
p + i
Print(Str(p) + " ")
EndIf
i + 2
Until p + i >= 1050
Input()
CloseConsole()

### Yabasic

sub isPrime(v)
if v < 2 then return False : fi
if mod(v, 2) = 0 then return v = 2 : fi
if mod(v, 3) = 0 then return v = 3 : fi
d = 5
while d * d <= v
if mod(v, d) = 0 then return False else d = d + 2 : fi
wend
return True
end sub

p = 3
i = 2

print "2 3 "; //special case
repeat
if isPrime(p + i) = 1 then
p = p + i
print p;
endif
i = i + 2
until p + i >= 1050
end

## C

#include <stdio.h>
#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;
}
}
}
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#)

// 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)

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

Here's another way of writing the mP sequence above which is (hopefully) a little clearer:

let mP = seq {
let mutable prevp, maxdiff = 2, 0
for p in primes32() do
let diff = p - prevp
if diff > maxdiff then
yield (prevp, diff, p)
maxdiff <- diff
prevp <- p
}

## Factor

Works with: Factor version 0.98
USING: formatting io kernel math math.primes ;

"2 " write 1 3
[ dup 1050 < ] [
2dup "(%d) %d " printf [ + next-prime ] keep 2dup - nip swap
] while 2drop nl
Output:
2 (1) 3 (2) 5 (6) 11 (8) 19 (10) 29 (12) 41 (18) 59 (20) 79 (22) 101 (26) 127 (30) 157 (34) 191 (36) 227 (42) 269 (44) 313 (46) 359 (50) 409 (52) 461 (60) 521 (66) 587 (72) 659 (74) 733 (76) 809 (78) 887 (80) 967 (82) 1049

## FreeBASIC

#include "isprime.bas"

dim as integer p = 3, i = 2

print "2 3 "; 'special case
do
if isprime( p + i ) then
p += i
print p;" ";
end if
i += 2
loop until p+i >=1050 : print
Output:
2 3  5  11  19  29  41  59  79  101  127  157  191  227  269  313  359  409  461  521  587  659  733  809  887  967  1049

## 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
}
}
}
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
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;
}
}
}
}
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

## jq

Works with: jq

Works with gojq, the Go implementation of jq

This entry uses `is_primes` as can be defined as in Erdős-primes#jq.

def primes:
2, (range(3;infinite;2) | select(is_prime));

def emit_until(cond; stream): label \$out | stream | if cond then break \$out else . end;

def special_primes:
foreach primes as \$p ({};
.emit = null
| if .p == null then .p = \$p | .emit = .p
else (\$p - .p) as \$g
| if \$g > .gap then .p = \$p | .gap=\$g | .emit = .p
else .
end
end;
select(.emit).emit);

# The following assumesg invocation with the -n option:
emit_until(. >= 1050; special_primes)
Output:

Invocation example: jq -n -f program.jq

2
3
5
11
19
29
41
59
79
101
127
157
191
227
269
313
359
409
461
521
587
659
733
809
887
967
1049

## Julia

using Primes

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

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

## Nim

import strutils, sugar

func isPrime(n: Positive): bool =
if (n and 1) == 0: return n == 2
var m = 3
while m * m <= n:
if n mod m == 0: return false
inc m, 2
result = true

iterator nextSpecialPrimes(lim: Positive): int =
assert lim >= 3
yield 2
yield 3
var last = 3
var lastGap = 1
for n in countup(5, lim, 2):
if not n.isPrime: continue
if n - last > lastGap:
lastGap = n - last
last = n
yield n

let list = collect(newSeq, for p in nextSpecialPrimes(1050): p)
echo "List of next special primes less than 1050:"
echo list.join(" ")
Output:
List of next special primes less than 1050:
2 3 5 11 19 29 41 59 79 101 127 157 191 227 269 313 359 409 461 521 587 659 733 809 887 967 1049

## Pascal

Library: primTrial

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

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.
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

## Perl

Library: ntheory
use strict;
use warnings;
use feature <state say>;
use ntheory 'primes';

my \$limit = 1050;

sub is_special {
state \$previous = 2;
state \$gap = 0;
state @primes = @{primes( 2*\$limit )};

shift @primes while \$primes[0] <= \$previous + \$gap;
\$gap = \$primes[0] - \$previous;
\$previous = \$primes[0];
[\$previous, \$gap];
}

my @specials = [2, 0];
do { push @specials, is_special() } until \$specials[-1][0] >= \$limit;

pop @specials;
printf "%4d %4d\n", @\$_ for @specials;
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

## 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

## Python

#!/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 = 3
i = 2

print("2 3", end = " ");
while True:
if isPrime(p + i) == 1:
p += i
print(p, end = " ");
i += 2
if p + i >= 1050:
break

## Raku

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);
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
/*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; sq.#= @.# **2 /*number of primes so far; prime². */
/* [↓] generate more primes ≤ high.*/
do [email protected].#+2 by 2 to hi /*find odd primes from here on. */
parse var j '' -1 _; if _==5 then iterate /*J ÷ by 5? (right digit).*/
if j//3==0 then iterate; if j//7==0 then iterate /*" " " 3? J ÷ by 7? */
do k=5 while sq.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; sq.#= j*j;  !.j= 1 /*bump # of Ps; assign next P; P²; P# */
end /*j*/; return
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

see "working..." + nl

Primes = []
limit1 = 100
oldPrime = 2

for n = 1 to limit1
nextPrime = oldPrime + n
if isprime(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

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...

## Sidef

func special_primes(upto) {

var gap = 0
var prev = 2
var list = [[prev, gap]]

loop {
var n = prev+gap
n = n.next_prime
break if (n > upto)
gap = n-prev
list << [n, gap]
prev = n
}

return list
}

special_primes(1050).each_2d {|p,gap|
say "#{'%4s' % p} #{'%4s' % gap}"
}
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

## Wren

Library: Wren-math
Library: Wren-fmt
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
}
}
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