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A341267 Gaps between first elements of quadruple primes of the form {p, p+2, p+6, p+12}.

%I A341267 #39 Feb 16 2025 08:34:01
%S A341267 6,6,24,60,126,120,294,450,186,150,54,6,120,1080,840,390,84,126,510,
%T A341267 150,144,3300,1230,870,1446,330,1794,726,1434,3360,1326,264,546,714,
%U A341267 1470,1836,1104,30,1026,204,336,744,2226,810,240,1050,270,1914,60,876,1980
%N A341267 Gaps between first elements of quadruple primes of the form {p, p+2, p+6, p+12}.
%C A341267 Primes in the quadruple need not be sequential primes.
%H A341267 Harvey P. Dale, <a href="/A341267/b341267.txt">Table of n, a(n) for n = 1..1000</a>
%H A341267 James S. DeArmon, <a href="https://github.com/JimDeArmon/forOEISdotorg/blob/main/gapQuadPrimes_v4.perl">Perl program</a>
%H A341267 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>
%F A341267 a(n) = A172454(n+1) - A172454(n).
%e A341267 The first 6 quadruples are (5,7,11,17), (11,13,17,23), (17,19,23,29), (41,43,47,53), (101,103,107,113), (227,229,233,239), so the first 5 terms of the sequence are 11-5=6, 17-11=6, 41-17=24, 101-41=60, 227-101=126.
%p A341267 b:= proc(n) option remember; local p; p:= `if`(n=1, 1, b(n-1));
%p A341267       do p:= nextprime(p);
%p A341267          if andmap(isprime, [p+2, p+6, p+12]) then return p fi
%p A341267       od
%p A341267     end:
%p A341267 a:= n-> b(n+1)-b(n):
%p A341267 seq(a(n), n=1..65);  # _Alois P. Heinz_, Feb 14 2021
%t A341267 Differences[Select[Prime[Range[5000]],AllTrue[#+{2,6,12},PrimeQ]&]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, May 21 2021 *)
%o A341267 (Perl) # See DeArmon link.
%Y A341267 Cf. A172454.
%K A341267 nonn
%O A341267 1,1
%A A341267 _James S. DeArmon_, Feb 07 2021