Prime triplets: Difference between revisions

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Task

Find and show members of prime triples (p, p+2, p+6), where p < 5500

See also

The OEIS entry: A022004 - Initial members of prime triples (p, p+2, p+6)
The Wikipedia entry: Prime triplet
The MathWorld entry: Prime Triplet
The RosettaCode task for just (p,p+4): Cousin primes
The RosettaCode task for other patterns of primes: Successive prime differences

11l

Translation of: Nim

F is_prime(n)
   I n == 2
      R 1B
   I n < 2 | n % 2 == 0
      R 0B
   L(i) (3 .. Int(sqrt(n))).step(2)
      I n % i == 0
         R 0B
   R 1B

print(‘   p  p+2  p+6’)
V count = 0
L(n) (3.<5500).step(2)
   I is_prime(n) & is_prime(n + 2) & is_prime(n + 6)
      print(f:‘{n:4} {n + 2:4} {n + 6:4}’)
      count++

print("\nFound "count‘ primes triplets for p < 5500.’)

Output:

   p  p+2  p+6
   5    7   11
  11   13   17
  17   19   23
  41   43   47
 101  103  107
 107  109  113
 191  193  197
 227  229  233
 311  313  317
 347  349  353
 461  463  467
 641  643  647
 821  823  827
 857  859  863
 881  883  887
1091 1093 1097
1277 1279 1283
1301 1303 1307
1427 1429 1433
1481 1483 1487
1487 1489 1493
1607 1609 1613
1871 1873 1877
1997 1999 2003
2081 2083 2087
2237 2239 2243
2267 2269 2273
2657 2659 2663
2687 2689 2693
3251 3253 3257
3461 3463 3467
3527 3529 3533
3671 3673 3677
3917 3919 3923
4001 4003 4007
4127 4129 4133
4517 4519 4523
4637 4639 4643
4787 4789 4793
4931 4933 4937
4967 4969 4973
5231 5233 5237
5477 5479 5483

Found 43 primes triplets for p < 5500.

Action!

Library: Action! Sieve of Eratosthenes

INCLUDE "H6:SIEVE.ACT"

PROC Main()
  DEFINE MAX="5499"
  BYTE ARRAY primes(MAX+1)
  INT i,count=[0]

  Put(125) PutE()
  Sieve(primes,MAX+1)
  FOR i=2 TO MAX-6
  DO
    IF primes(i)=1 AND primes(i+2)=1 AND (primes(i+6))=1 THEN
      PrintF("(%I %I %I) ",i,i+2,i+6)
      count==+1
    FI
  OD
  PrintF("%E%EThere are %I prime triplets",count)
RETURN

Output:

Screenshot from Atari 8-bit computer

(5 7 11) (11 13 17) (17 19 23) (41 43 47) (101 103 107) (107 109 113) (191 193 197) (227 229 233) (311 313 317) (347 349 353)
(461 463 467) (641 643 647) (821 823 827) (857 859 863) (881 883 887) (1091 1093 1097) (1277 1279 1283) (1301 1303 1307)
(1427 1429 1433) (1481 1483 1487) (1487 1489 1493) (1607 1609 1613) (1871 1873 1877) (1997 1999 2003) (2081 2083 2087)
(2237 2239 2243) (2267 2269 2273) (2657 2659 2663) (2687 2689 2693) (3251 3253 3257) (3461 3463 3467) (3527 3529 3533)
(3671 3673 3677) (3917 3919 3923) (4001 4003 4007) (4127 4129 4133) (4517 4519 4523) (4637 4639 4643) (4787 4789 4793)
(4931 4933 4937) (4967 4969 4973) (5231 5233 5237) (5477 5479 5483)

There are 43 prime triplets

ALGOL 68

Using code from Successive_prime_differences#ALGOL_68

BEGIN # find primes p where p+2 and p+6 are also prime                    #
    # reurns a list of primes up to n #
    PROC prime list = ( INT n )[]INT:
         BEGIN
            # sieve the primes to n #
            INT no = 0, yes = 1;
            [ 1 : n ]INT p;
            p[ 1 ] := no; p[ 2 ] := yes;
            FOR i FROM 3 BY 2 TO n DO p[ i ] := yes OD;
            FOR i FROM 4 BY 2 TO n DO p[ i ] := no  OD;
            FOR i FROM 3 BY 2 TO ENTIER sqrt( n ) DO
                IF p[ i ] = yes THEN FOR s FROM i * i BY i + i TO n DO p[ s ] := no OD FI
            OD;
            # replace the sieve with a list #
            INT p pos := 0;
            FOR i TO n DO IF p[ i ] = yes THEN p[ p pos +:= 1 ] := i FI OD;
            p[ 1 : p pos ]
         END # prime list # ;
    # prints the elements of list #
    PROC print list = ( INT width, []INT list )VOID:
         BEGIN
            print( ( "[" ) );
            FOR i FROM LWB list TO UPB list DO print( ( " ", whole( list[ i ], width ) ) ) OD;
            print( ( " ]" ) )
         END # print list # ;
    # attempts to find patterns in the differences of primes and prints the results #
    PROC try differences = ( []INT primes, []INT pattern )VOID:
         BEGIN
            INT    pattern length = ( UPB pattern - LWB pattern ) + 1;
            [ 1 :  pattern length + 1 ]INT first; FOR i TO UPB first DO first[ i ] := 0 OD;
            [ 1 :  pattern length + 1 ]INT last;  FOR i TO UPB last  DO last[  i ] := 0 OD;
            INT    count := 0;
            FOR p FROM LWB primes + pattern length TO UPB primes DO
                BOOL matched := TRUE;
                INT e pos    := LWB pattern;
                FOR e FROM p - pattern length TO p - 1
                WHILE matched := primes[ e + 1 ] - primes[ e ] = pattern[ e pos ]
                DO
                    e pos +:= 1
                OD;
                IF matched THEN
                    # found a matching sequence #
                    count +:= 1;
                    print list( -4, primes[ p - pattern length : p ] );
                    IF count MOD 6 = 0 THEN print( ( newline ) ) ELSE print( ( " " ) ) FI
                FI
            OD;
            print( ( newline, "Found ", whole( count, 0 ), " prime sequence(s) that differ by: " ) );
            print list( 0, pattern );
            print( ( newline ) )
        END # try differences # ;
    INT max number = 5 500;
    []INT    p list = prime list( max number - 1 );
    print( ( "Prime triplets under ", whole( max number, 0 ), ":", newline ) );
    try differences( p list, ( 2, 4 ) )
END

Output:

Prime triplets under 5500:
[    5    7   11 ] [   11   13   17 ] [   17   19   23 ] [   41   43   47 ] [  101  103  107 ] [  107  109  113 ]
[  191  193  197 ] [  227  229  233 ] [  311  313  317 ] [  347  349  353 ] [  461  463  467 ] [  641  643  647 ]
[  821  823  827 ] [  857  859  863 ] [  881  883  887 ] [ 1091 1093 1097 ] [ 1277 1279 1283 ] [ 1301 1303 1307 ]
[ 1427 1429 1433 ] [ 1481 1483 1487 ] [ 1487 1489 1493 ] [ 1607 1609 1613 ] [ 1871 1873 1877 ] [ 1997 1999 2003 ]
[ 2081 2083 2087 ] [ 2237 2239 2243 ] [ 2267 2269 2273 ] [ 2657 2659 2663 ] [ 2687 2689 2693 ] [ 3251 3253 3257 ]
[ 3461 3463 3467 ] [ 3527 3529 3533 ] [ 3671 3673 3677 ] [ 3917 3919 3923 ] [ 4001 4003 4007 ] [ 4127 4129 4133 ]
[ 4517 4519 4523 ] [ 4637 4639 4643 ] [ 4787 4789 4793 ] [ 4931 4933 4937 ] [ 4967 4969 4973 ] [ 5231 5233 5237 ]
[ 5477 5479 5483 ]
Found 43 prime sequence(s) that differ by: [ 2 4 ]

Arturo

lst: select select 2..5500 => prime? 'x 
        -> and? [prime? x+2] [prime? x+6]

loop split.every: 5 lst 'a -> 
    print map a 'item [
        pad join.with:", " 
            to [:string] @[item item+2 item+6] 17
    ]

Output:

         5, 7, 11        11, 13, 17        17, 19, 23        41, 43, 47     101, 103, 107 
    107, 109, 113     191, 193, 197     227, 229, 233     311, 313, 317     347, 349, 353 
    461, 463, 467     641, 643, 647     821, 823, 827     857, 859, 863     881, 883, 887 
 1091, 1093, 1097  1277, 1279, 1283  1301, 1303, 1307  1427, 1429, 1433  1481, 1483, 1487 
 1487, 1489, 1493  1607, 1609, 1613  1871, 1873, 1877  1997, 1999, 2003  2081, 2083, 2087 
 2237, 2239, 2243  2267, 2269, 2273  2657, 2659, 2663  2687, 2689, 2693  3251, 3253, 3257 
 3461, 3463, 3467  3527, 3529, 3533  3671, 3673, 3677  3917, 3919, 3923  4001, 4003, 4007 
 4127, 4129, 4133  4517, 4519, 4523  4637, 4639, 4643  4787, 4789, 4793  4931, 4933, 4937 
 4967, 4969, 4973  5231, 5233, 5237  5477, 5479, 5483

AWK

# syntax: GAWK -f PRIME_TRIPLETS.AWK
BEGIN {
    start = 1
    stop = 5499
    for (i=start; i<=stop; i++) {
      if (is_prime(i+6) && is_prime(i+2) && is_prime(i)) {
        printf("%d %d %d\n",i,i+2,i+6)
        count++
      }
    }
    printf("Prime Triplets %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:

5 7 11
11 13 17
17 19 23
41 43 47
101 103 107
107 109 113
191 193 197
227 229 233
311 313 317
347 349 353
461 463 467
641 643 647
821 823 827
857 859 863
881 883 887
1091 1093 1097
1277 1279 1283
1301 1303 1307
1427 1429 1433
1481 1483 1487
1487 1489 1493
1607 1609 1613
1871 1873 1877
1997 1999 2003
2081 2083 2087
2237 2239 2243
2267 2269 2273
2657 2659 2663
2687 2689 2693
3251 3253 3257
3461 3463 3467
3527 3529 3533
3671 3673 3677
3917 3919 3923
4001 4003 4007
4127 4129 4133
4517 4519 4523
4637 4639 4643
4787 4789 4793
4931 4933 4937
4967 4969 4973
5231 5233 5237
5477 5479 5483
Prime Triplets 1-5499: 43

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

for p = 3 to 5499 step 2
	if not isPrime(p+6) then continue for
	if not isPrime(p+2) then continue for
	if not isPrime(p) then continue for
	print "["; p; " "; p+2; " "; p+6; "]"
next p
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()
For p.i = 3 To 5499 Step 2
  If Not isPrime(p+6) 
    Continue
  EndIf
  If Not isPrime(p+2) 
    Continue
  EndIf
  If Not isPrime(p) 
    Continue
  EndIf        
  PrintN("["+ Str(p) + " " + Str(p+2) + " " + Str(p+6) + "]")
Next p
PrintN(#CRLF$ + "--- terminado, pulsa RETURN---"): 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

for p = 3 to 5499 step 2
    if not isPrime(p+6) continue
    if not isPrime(p+2) continue
    if not isPrime(p) continue
    print "[", p using "####", p+2 using "####", p+6 using "####", "]"
next p
end

Factor

Works with: Factor version 0.98

USING: arrays kernel lists lists.lazy math math.primes
math.primes.lists prettyprint sequences ;

lprimes                                ! An infinite lazy list of primes
[ dup 2 + dup 4 + 3array ] lmap-lazy   ! Map primes to their triplets (e.g. 2 -> { 2 4 8 })
[ [ prime? ] all? ] lfilter            ! Select triplets which contain only primes
[ first 5500 < ] lwhile                ! Make the list end eventually...
[ . ] leach                            ! Print each item in the list

Output:

{ 5 7 11 }
{ 11 13 17 }
{ 17 19 23 }
{ 41 43 47 }
{ 101 103 107 }
{ 107 109 113 }
{ 191 193 197 }
{ 227 229 233 }
{ 311 313 317 }
{ 347 349 353 }
{ 461 463 467 }
{ 641 643 647 }
{ 821 823 827 }
{ 857 859 863 }
{ 881 883 887 }
{ 1091 1093 1097 }
{ 1277 1279 1283 }
{ 1301 1303 1307 }
{ 1427 1429 1433 }
{ 1481 1483 1487 }
{ 1487 1489 1493 }
{ 1607 1609 1613 }
{ 1871 1873 1877 }
{ 1997 1999 2003 }
{ 2081 2083 2087 }
{ 2237 2239 2243 }
{ 2267 2269 2273 }
{ 2657 2659 2663 }
{ 2687 2689 2693 }
{ 3251 3253 3257 }
{ 3461 3463 3467 }
{ 3527 3529 3533 }
{ 3671 3673 3677 }
{ 3917 3919 3923 }
{ 4001 4003 4007 }
{ 4127 4129 4133 }
{ 4517 4519 4523 }
{ 4637 4639 4643 }
{ 4787 4789 4793 }
{ 4931 4933 4937 }
{ 4967 4969 4973 }
{ 5231 5233 5237 }
{ 5477 5479 5483 }

Fermat

for i=3,5499,2 do if Isprime(i)=1 and Isprime(i+2)=1 and Isprime(i+6)=1 then !!(i,i+2,i+6) fi od

FreeBASIC

#include "isprime.bas"
for p as uinteger = 3 to 5499 step 2
     if not isprime(p+6) then continue for
     if not isprime(p+2) then continue for
     if not isprime(p) then continue for
     print using "[#### #### ####] ";p;p+2;p+6;
next p

Output:


[   5    7   11] [  11   13   17] [  17   19   23] [  41   43   47] [ 101  103  107] [ 107  109  113] [ 191  193  197] [ 227  229  233] [ 311  313  317] [ 347  349  353] [ 461  463  467] [ 641  643  647] [ 821  823  827] [ 857  859  863] [ 881  883  887] [1091 1093 1097] [1277 1279 1283] [1301 1303 1307] [1427 1429 1433] [1481 1483 1487] [1487 1489 1493] [1607 1609 1613] [1871 1873 1877] [1997 1999 2003] [2081 2083 2087] [2237 2239 2243] [2267 2269 2273] [2657 2659 2663] [2687 2689 2693] [3251 3253 3257] [3461 3463 3467] [3527 3529 3533] [3671 3673 3677] [3917 3919 3923] [4001 4003 4007] [4127 4129 4133] [4517 4519 4523] [4637 4639 4643] [4787 4789 4793] [4931 4933 4937] [4967 4969 4973] [5231 5233 5237] [5477 5479 5483]

Go

Translation of: Wren

Library: Go-rcu

package main

import (
    "fmt"
    "rcu"
)

func main() {
    c := rcu.PrimeSieve(5505, false)
    var triples [][3]int
    fmt.Println("Prime triplets: p, p + 2, p + 6 where p < 5,500:")
    for i := 3; i < 5500; i += 2 {
        if !c[i] && !c[i+2] && !c[i+6] {
            triples = append(triples, [3]int{i, i + 2, i + 6})
        }
    }
    for _, triple := range triples {
        var t [3]string
        for i := 0; i < 3; i++ {
            t[i] = rcu.Commatize(triple[i])
        }
        fmt.Printf("%5s  %5s  %5s\n", t[0], t[1], t[2])
    }
    fmt.Println("\nFound", len(triples), "such prime triplets.")
}

Output:

Same as Wren entry.

GW-BASIC

10 FOR A = 3 TO 5499 STEP 2
20 P = A
30 GOSUB 1000
40 IF Z = 0 THEN GOTO 500
50 P = A + 2
60 GOSUB 1000
70 IF Z = 0 THEN GOTO 500
80 P = A + 6
90 GOSUB 1000
100 IF Z = 1 THEN PRINT A,A+2,A+6
500 NEXT A
510 END
1000 Z = 1 : I = 2
1010 IF P MOD I = 0 THEN Z = 0 : RETURN
1020 I = I + 1
1030 IF I*I > P THEN RETURN
1040 GOTO 1010

jq

Works with: jq

Works with gojq, the Go implementation of jq

The implementation of `is_prime` at Erdős-primes#jq can be used here and is therefore not repeated.

The `prime_triplets` defined here generates an unbounded stream of prime triples, which is harnessed by the generic function `emit_until` defined as follows:

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

The Task:

# Output: [p,p+2,p+6] where p is prime
def prime_triplets:
  def pt: .[2] == .[1] + 4 and .[1] == .[0] + 2;
  def next: .[1:] + [first( range(.[2] + 2; infinite;2) | select(is_prime))];
  # prime the foreach with the first triplet
  foreach range(7; infinite; 2) as $i ([2,3,5]; next; select(pt) ) ;

emit_until(.[0] >= 5500; prime_triplets)

Output:

[5,7,11]
[11,13,17]
[17,19,23]
[41,43,47]
[101,103,107]
[107,109,113]
[191,193,197]
[227,229,233]
[311,313,317]
[347,349,353]
[461,463,467]
[641,643,647]
[821,823,827]
[857,859,863]
[881,883,887]
[1091,1093,1097]
[1277,1279,1283]
[1301,1303,1307]
[1427,1429,1433]
[1481,1483,1487]
[1487,1489,1493]
[1607,1609,1613]
[1871,1873,1877]
[1997,1999,2003]
[2081,2083,2087]
[2237,2239,2243]
[2267,2269,2273]
[2657,2659,2663]
[2687,2689,2693]
[3251,3253,3257]
[3461,3463,3467]
[3527,3529,3533]
[3671,3673,3677]
[3917,3919,3923]
[4001,4003,4007]
[4127,4129,4133]
[4517,4519,4523]
[4637,4639,4643]
[4787,4789,4793]
[4931,4933,4937]
[4967,4969,4973]
[5231,5233,5237]
[5477,5479,5483]

Julia

using Primes

pmask = primesmask(1, 5505)
foreach(n -> println([n, n + 2, n + 6]), filter(n -> pmask[n] && pmask[n + 2] && pmask[n + 6], 1:5500))

Output:

[5, 7, 11]
[11, 13, 17]
[17, 19, 23]
[41, 43, 47]
[101, 103, 107]
[107, 109, 113]
[191, 193, 197]
[227, 229, 233]
[311, 313, 317]
[347, 349, 353]
[461, 463, 467]
[641, 643, 647]
[821, 823, 827]
[857, 859, 863]
[881, 883, 887]
[1091, 1093, 1097]
[1277, 1279, 1283]
[1301, 1303, 1307]
[1427, 1429, 1433]
[1481, 1483, 1487]
[1487, 1489, 1493]
[1607, 1609, 1613]
[1871, 1873, 1877]
[1997, 1999, 2003]
[2081, 2083, 2087]
[2237, 2239, 2243]
[2267, 2269, 2273]
[2657, 2659, 2663]
[2687, 2689, 2693]
[3251, 3253, 3257]
[3461, 3463, 3467]
[3527, 3529, 3533]
[3671, 3673, 3677]
[3917, 3919, 3923]
[4001, 4003, 4007]
[4127, 4129, 4133]
[4517, 4519, 4523]
[4637, 4639, 4643]
[4787, 4789, 4793]
[4931, 4933, 4937]
[4967, 4969, 4973]
[5231, 5233, 5237]
[5477, 5479, 5483]

Mathematica / Wolfram Language

Cases[Partition[Most@NestWhileList[NextPrime, 2, # < 5500 &], 3, 
   1], _?(Differences[#] == {2, 4} &)] // TableForm

Output:

5 7 11 11 13 17 17 19 23 41 43 47 101 103 107 107 109 113 191 193 197 227 229 233 311 313 317 347 349 353 461 463 467 641 643 647 821 823 827 857 859 863 881 883 887 1091 1093 1097 1277 1279 1283 1301 1303 1307 1427 1429 1433 1481 1483 1487 1487 1489 1493 1607 1609 1613 1871 1873 1877 1997 1999 2003 2081 2083 2087 2237 2239 2243 2267 2269 2273 2657 2659 2663 2687 2689 2693 3251 3253 3257 3461 3463 3467 3527 3529 3533 3671 3673 3677 3917 3919 3923 4001 4003 4007 4127 4129 4133 4517 4519 4523 4637 4639 4643 4787 4789 4793 4931 4933 4937 4967 4969 4973 5231 5233 5237 5477 5479 5483

Nim

import strformat

const
  N = 5500 - 1
  Max = N + 6

# Sieve of Erathosthenes: false (default) is composite.
var composite: array[3..Max, bool]   # Ignore 2 as all primes should be odd.
var n = 3
while true:
  let n2 = n * n
  if n2 > Max: break
  if not composite[n]:
    for k in countup(n2, Max, 2 * n):
      composite[k] = true
  inc n, 2

template isPrime(n: int): bool = not composite[n]

echo "   p  p+2  p+6"
var count = 0
for n in countup(3, N, 2):
  if n.isPrime and (n + 2).isPrime and (n + 6).isPrime:
    echo &"{n:4} {n+2:4} {n+6:4}"
    inc count

echo &"\nFound {count} primes triplets for p < {N+1}."

Output:

   p  p+2  p+6
   5    7   11
  11   13   17
  17   19   23
  41   43   47
 101  103  107
 107  109  113
 191  193  197
 227  229  233
 311  313  317
 347  349  353
 461  463  467
 641  643  647
 821  823  827
 857  859  863
 881  883  887
1091 1093 1097
1277 1279 1283
1301 1303 1307
1427 1429 1433
1481 1483 1487
1487 1489 1493
1607 1609 1613
1871 1873 1877
1997 1999 2003
2081 2083 2087
2237 2239 2243
2267 2269 2273
2657 2659 2663
2687 2689 2693
3251 3253 3257
3461 3463 3467
3527 3529 3533
3671 3673 3677
3917 3919 3923
4001 4003 4007
4127 4129 4133
4517 4519 4523
4637 4639 4643
4787 4789 4793
4931 4933 4937
4967 4969 4973
5231 5233 5237
5477 5479 5483

Found 43 primes triplets for p < 5500.

PARI/GP

for(i=1,5499,if(isprime(i)&&isprime(i+2)&&isprime(i+6),print(i," ",i+2," ",i+6)))

Perl

Library: ntheory

#!/usr/bin/perl

use strict;
use warnings;
use ntheory qw( is_prime twin_primes );

is_prime($_ + 6) and printf "%5d" x 3 . "\n", $_, $_ + 2, $_ + 6
  for @{ twin_primes( 5500 ) };

Output:

    5    7   11
   11   13   17
   17   19   23
   41   43   47
  101  103  107
  107  109  113
  191  193  197
  227  229  233
  311  313  317
  347  349  353
  461  463  467
  641  643  647
  821  823  827
  857  859  863
  881  883  887
 1091 1093 1097
 1277 1279 1283
 1301 1303 1307
 1427 1429 1433
 1481 1483 1487
 1487 1489 1493
 1607 1609 1613
 1871 1873 1877
 1997 1999 2003
 2081 2083 2087
 2237 2239 2243
 2267 2269 2273
 2657 2659 2663
 2687 2689 2693
 3251 3253 3257
 3461 3463 3467
 3527 3529 3533
 3671 3673 3677
 3917 3919 3923
 4001 4003 4007
 4127 4129 4133
 4517 4519 4523
 4637 4639 4643
 4787 4789 4793
 4931 4933 4937
 4967 4969 4973
 5231 5233 5237
 5477 5479 5483

Phix

function pt(integer p) return is_prime(p+2) and is_prime(p+6) end function
sequence res = filter(get_primes_le(5500),pt)
         res = apply(true,sq_add,{res,{{0,2,6}}})
         res = apply(true,sprintf,{{"(%d %d %d)"},res})
printf(1,"Found %d prime triplets: %s\n",{length(res),join(shorten(res,"",2),", ")})

Output:

Found 43 prime triplets: (5 7 11), (11 13 17), ..., (5231 5233 5237), (5477 5479 5483)

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__':
    for p in range(3, 5499, 2):
        if not isPrime(p+6):
            continue
        if not isPrime(p+2):
            continue
        if not isPrime(p):
            continue
        print(f'[{p} {p+2} {p+6}]')

Raku

Adapted from Cousin primes

Filter

Favoring brevity over efficiency due to the small range of n, the most concise solution is:

say grep *.all.is-prime, map { $_, $_+2, $_+6 }, 2..5500;

Output:

((5 7 11) (11 13 17) (17 19 23) (41 43 47) (101 103 107) (107 109 113) (191 193 197) (227 229 233) (311 313 317) (347 349 353) (461 463 467) (641 643 647) (821 823 827) (857 859 863) (881 883 887) (1091 1093 1097) (1277 1279 1283) (1301 1303 1307) (1427 1429 1433) (1481 1483 1487) (1487 1489 1493) (1607 1609 1613) (1871 1873 1877) (1997 1999 2003) (2081 2083 2087) (2237 2239 2243) (2267 2269 2273) (2657 2659 2663) (2687 2689 2693) (3251 3253 3257) (3461 3463 3467) (3527 3529 3533) (3671 3673 3677) (3917 3919 3923) (4001 4003 4007) (4127 4129 4133) (4517 4519 4523) (4637 4639 4643) (4787 4789 4793) (4931 4933 4937) (4967 4969 4973) (5231 5233 5237) (5477 5479 5483))

Infinite List

A more efficient and versatile approach is to generate an infinite list of triple primes, using this info from https://oeis.org/A022004 :

All terms are congruent to 5 (mod 6).

constant @triples = (5, *+6 … *).map: -> \n { $_ if .all.is-prime given (n, n+2, n+6) }

my $count = @triples.first: :k, *.[0] >= 5500;

say .fmt('%4d') for @triples.head($count);

Output:

   5    7   11
  11   13   17
  17   19   23
  41   43   47
 101  103  107
 107  109  113
 191  193  197
 227  229  233
 311  313  317
 347  349  353
 461  463  467
 641  643  647
 821  823  827
 857  859  863
 881  883  887
1091 1093 1097
1277 1279 1283
1301 1303 1307
1427 1429 1433
1481 1483 1487
1487 1489 1493
1607 1609 1613
1871 1873 1877
1997 1999 2003
2081 2083 2087
2237 2239 2243
2267 2269 2273
2657 2659 2663
2687 2689 2693
3251 3253 3257
3461 3463 3467
3527 3529 3533
3671 3673 3677
3917 3919 3923
4001 4003 4007
4127 4129 4133
4517 4519 4523
4637 4639 4643
4787 4789 4793
4931 4933 4937
4967 4969 4973
5231 5233 5237
5477 5479 5483

REXX

/*REXX program finds prime triplets:  P, P+2, P+6  are primes, and  P < some specified N*/
parse arg hi cols .                              /*obtain optional argument from the CL.*/
if   hi=='' |   hi==","  then   hi= 5500         /*Not specified?  Then use the default.*/
if cols=='' | cols==","  then cols=    4         /* "      "         "   "   "     "    */
call genP hi + 6                                 /*build semaphore array for low primes.*/
     do p=1  while @.p<hi
     end  /*p*/;           lim= p-1              /*set  LIM   to the  Pth  prime.       */
w= 30                                            /*width of a prime triplet in a column.*/
__= ' ';       @trip= ' prime triplets:  p, p+2, p+6  are primes,  and p  < '   commas(hi)
if cols>0 then say ' index │'center(@trip,   1 + cols*(w+1)     )
if cols>0 then say '───────┼'center(""   ,   1 + cols*(w+1), '─')
Tprimes= 0;                idx= 1                /*initialize # prime triplets  & index.*/
$=                                               /*a list of  prime triplets  (so far). */
     do j=1  to  lim                             /*look for prime triplets within range.*/
     p2= @.j + 2;  if \!.p2  then iterate        /*is  P2  prime?    No, then skip it.  */
     p6= p2  + 4;  if \!.p6  then iterate        /* "  P6    "        "    "    "   "   */
     Tprimes= Tprimes + 1                        /*bump the number of  prime triplets.  */
     if cols==0              then iterate        /*Build the list  (to be shown later)? */
     @@@= commas(@.j)__ commas(p2)__ commas(p6)  /*add commas & blanks to prime triplet.*/
     $= $ left( '('@@@")",  w)                   /*add a prime triplet ──► the  $  list.*/
     if Tprimes//cols\==0    then iterate        /*have we populated a line of output?  */
     say center(idx, 7)'│' strip(substr($, 2), 'T');    $=   /*show what we have so far.*/
     idx= idx + cols                             /*bump the  index  count for the output*/
     end   /*j*/

if $\==''  then say center(idx, 7)"│" strip(substr($, 2), 'T')  /*possible show residual*/
if cols>0 then say '───────┴'center(""                         ,  1 + cols*(w+1), '─')
say
say 'Found '       commas(Tprimes)      @trip
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;            parse arg limit          /*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 limit               /*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  3  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

output when using the default inputs:

 index │                                  prime triplets:  p, p+2, p+6  are primes,  and p  <  5,500
───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1   │ (5  7  11)                     (11  13  17)                   (17  19  23)                   (41  43  47)
   5   │ (101  103  107)                (107  109  113)                (191  193  197)                (227  229  233)
   9   │ (311  313  317)                (347  349  353)                (461  463  467)                (641  643  647)
  13   │ (821  823  827)                (857  859  863)                (881  883  887)                (1,091  1,093  1,097)
  17   │ (1,277  1,279  1,283)          (1,301  1,303  1,307)          (1,427  1,429  1,433)          (1,481  1,483  1,487)
  21   │ (1,487  1,489  1,493)          (1,607  1,609  1,613)          (1,871  1,873  1,877)          (1,997  1,999  2,003)
  25   │ (2,081  2,083  2,087)          (2,237  2,239  2,243)          (2,267  2,269  2,273)          (2,657  2,659  2,663)
  29   │ (2,687  2,689  2,693)          (3,251  3,253  3,257)          (3,461  3,463  3,467)          (3,527  3,529  3,533)
  33   │ (3,671  3,673  3,677)          (3,917  3,919  3,923)          (4,001  4,003  4,007)          (4,127  4,129  4,133)
  37   │ (4,517  4,519  4,523)          (4,637  4,639  4,643)          (4,787  4,789  4,793)          (4,931  4,933  4,937)
  41   │ (4,967  4,969  4,973)          (5,231  5,233  5,237)          (5,477  5,479  5,483)
───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  43  prime triplets:  p, p+2, p+6  are primes,  and p  <  5,500

Ring

load "stdlib.ring"
see "working..." + nl
see "Initial members of prime triples (p, p+2, p+6) are:" + nl
see "p p+2 p+6" + nl
row = 0
limit = 5500
 
for n = 1 to limit
    if isprime(n) and isprime(n+2) and isprime(n+6)
       row = row + 1
       see "" + n + " " + (n+2) + " " + (n+6) + nl
    ok
next

see "Found " + row + " primes" + nl
see "done..." + nl

Output:

working...
Initial members of prime triples (p, p+2, p+6) are:
p p+2 p+6
5 7 11
11 13 17
17 19 23
41 43 47
101 103 107
107 109 113
191 193 197
227 229 233
311 313 317
347 349 353
461 463 467
641 643 647
821 823 827
857 859 863
881 883 887
1091 1093 1097
1277 1279 1283
1301 1303 1307
1427 1429 1433
1481 1483 1487
1487 1489 1493
1607 1609 1613
1871 1873 1877
1997 1999 2003
2081 2083 2087
2237 2239 2243
2267 2269 2273
2657 2659 2663
2687 2689 2693
3251 3253 3257
3461 3463 3467
3527 3529 3533
3671 3673 3677
3917 3919 3923
4001 4003 4007
4127 4129 4133
4517 4519 4523
4637 4639 4643
4787 4789 4793
4931 4933 4937
4967 4969 4973
5231 5233 5237
5477 5479 5483
Found 43 primes
done...

Sidef

say "Values of p such that (p, p+2, p+6) are all prime:"
5500.primes.grep{|p| all_prime(p+2, p+6) }.say

Output:

Values of p such that (p, p+2, p+6) are all prime:
[5, 11, 17, 41, 101, 107, 191, 227, 311, 347, 461, 641, 821, 857, 881, 1091, 1277, 1301, 1427, 1481, 1487, 1607, 1871, 1997, 2081, 2237, 2267, 2657, 2687, 3251, 3461, 3527, 3671, 3917, 4001, 4127, 4517, 4637, 4787, 4931, 4967, 5231, 5477]

Tiny BASIC

    LET A = 1
 10 LET A = A + 2
    IF A > 5499 THEN END
    LET P = A
    GOSUB 100
    IF Z = 0 THEN GOTO 10
    LET P = A + 2
    GOSUB 100
    IF Z = 0 THEN GOTO 10
    LET P = A + 6
    GOSUB 100
    IF Z = 0 THEN GOTO 10
    PRINT A," ",A+2," ",A+6
    GOTO 10
100 REM PRIMALITY BY TRIAL DIVISION
    LET Z = 1
    LET I = 2
110 IF (P/I)*I = P THEN LET Z = 0
    IF Z = 0 THEN RETURN
    LET I = I + 1
    IF I*I <= P THEN GOTO 110
    RETURN

Wren

Library: Wren-math

Library: Wren-fmt

import "/math" for Int
import "/fmt" for Fmt

var c = Int.primeSieve(5505, false)
var triples = []
System.print("Prime triplets: p, p + 2, p + 6 where p < 5,500:")
var i = 3
while (i < 5500) {
    if (!c[i] && !c[i+2] && !c[i+6]) triples.add([i, i+2, i+6])
    i = i + 2
}
for (triple in triples) Fmt.print("$,6d", triple)
System.print("\nFound %(triples.count) such prime triplets.")

Output:

Prime triplets: p, p + 2, p + 6 where p < 5,500:
     5      7     11
    11     13     17
    17     19     23
    41     43     47
   101    103    107
   107    109    113
   191    193    197
   227    229    233
   311    313    317
   347    349    353
   461    463    467
   641    643    647
   821    823    827
   857    859    863
   881    883    887
 1,091  1,093  1,097
 1,277  1,279  1,283
 1,301  1,303  1,307
 1,427  1,429  1,433
 1,481  1,483  1,487
 1,487  1,489  1,493
 1,607  1,609  1,613
 1,871  1,873  1,877
 1,997  1,999  2,003
 2,081  2,083  2,087
 2,237  2,239  2,243
 2,267  2,269  2,273
 2,657  2,659  2,663
 2,687  2,689  2,693
 3,251  3,253  3,257
 3,461  3,463  3,467
 3,527  3,529  3,533
 3,671  3,673  3,677
 3,917  3,919  3,923
 4,001  4,003  4,007
 4,127  4,129  4,133
 4,517  4,519  4,523
 4,637  4,639  4,643
 4,787  4,789  4,793
 4,931  4,933  4,937
 4,967  4,969  4,973
 5,231  5,233  5,237
 5,477  5,479  5,483

Found 43 such prime triplets.

XPL0

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

int Count, P;
[ChOut(0, ^ );
Count:= 0;
P:= 3;
repeat  if IsPrime(P) & IsPrime(P+2) & IsPrime(P+6) then
            [IntOut(0, P);  ChOut(0, ^ );
            IntOut(0, P+2);  ChOut(0, ^ );
            IntOut(0, P+6);  ChOut(0, ^ );
            Count:= Count+1;
            if rem(Count/5) then ChOut(0, 9\tab\) else CrLf(0);
            ];
        P:= P+2;
until   P >= 5500;
CrLf(0);
IntOut(0, Count);
Text(0, " prime triplets found below 5500.
");
]

Output:

 5 7 11         11 13 17        17 19 23        41 43 47        101 103 107 
107 109 113     191 193 197     227 229 233     311 313 317     347 349 353 
461 463 467     641 643 647     821 823 827     857 859 863     881 883 887 
1091 1093 1097  1277 1279 1283  1301 1303 1307  1427 1429 1433  1481 1483 1487 
1487 1489 1493  1607 1609 1613  1871 1873 1877  1997 1999 2003  2081 2083 2087 
2237 2239 2243  2267 2269 2273  2657 2659 2663  2687 2689 2693  3251 3253 3257 
3461 3463 3467  3527 3529 3533  3671 3673 3677  3917 3919 3923  4001 4003 4007 
4127 4129 4133  4517 4519 4523  4637 4639 4643  4787 4789 4793  4931 4933 4937 
4967 4969 4973  5231 5233 5237  5477 5479 5483  
43 prime triplets found below 5500.