A342502 - OEIS

%I #24 May 14 2022 11:21:57

%S 12,210,1050,330,1920,390,720,150,22950,10710,780,5040,27060,26040,

%T 2340,13440,8880,360,1950,41370,17790,3630,4320,6510,870,76620,15210,

%U 21540,5760,29100,2340,66990,1950,3360,5370,16800,6930,40530,4230,3570,15510,10320

%N Gaps between first elements of prime quintuples of the form (p, p+2, p+6, p+12, p+14). The quintuples are abutting: twin/cousin/sexy/twin pairs.

%H T. Forbes, <a href="http://anthony.d.forbes.googlepages.com/adf.htm">Prime k-tuplets</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>.

%F a(n) = A078946(n) - A078946(n-1) for n >= 2.

%F a(n) == 0 (mod 30) for n>1.

%e The first 4 terms of the sequence are 12, 210, 1050, 330, since the gaps between first elements of the first five quintuples {5,7,11,17,19}, {17,19,23,29,31}, {227,229,233,239,241}, {1277,1279,1283,1289,1291}, {1607,1609,1613,1619,1621} are, 17-5=12, 227-17=210, etc.

%p b:= proc(n) option remember; local p; p:= `if`(n=1, 1, b(n-1));

%p       do p:= nextprime(p);

%p          if andmap(isprime, [p+2, p+6, p+12, p+14]) then return p fi

%p       od

%p     end:

%p a:= n-> b(n+1)-b(n):

%p seq(a(n), n=1..65);  # _Alois P. Heinz_, Mar 14 2021

%t b[n_] := b[n] = Module[{p}, p = If[n == 1, 1, b[n-1]]; While[True, p = NextPrime[p]; If[AllTrue[{p+2, p+6, p+12, p+14}, PrimeQ], Return[p]]]];

%t a[n_] := b[n+1]-b[n];

%t Table[a[n], {n, 1, 65}] (* _Jean-François Alcover_, May 14 2022, after _Alois P. Heinz_ *)

%Y Cf. A078946.

%K nonn

%O 1,1

%A _James S. DeArmon_, Mar 14 2021