A342502 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.

12, 210, 1050, 330, 1920, 390, 720, 150, 22950, 10710, 780, 5040, 27060, 26040, 2340, 13440, 8880, 360, 1950, 41370, 17790, 3630, 4320, 6510, 870, 76620, 15210, 21540, 5760, 29100, 2340, 66990, 1950, 3360, 5370, 16800, 6930, 40530, 4230, 3570, 15510, 10320

Offset: 1

Author

James S. DeArmon, Mar 14 2021

Keywords

• nonn

Examples

			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.
		

Links

• T. Forbes, Prime k-tuplets
• Eric Weisstein's World of Mathematics, Prime Triplet.

Crossrefs

Cf. A078946.

Programs

Maple

b:= proc(n) option remember; local p; p:= `if`(n=1, 1, b(n-1));
      do p:= nextprime(p);
         if andmap(isprime, [p+2, p+6, p+12, p+14]) then return p fi
      od
    end:
a:= n-> b(n+1)-b(n):
seq(a(n), n=1..65);  # Alois P. Heinz, Mar 14 2021

Mathematica

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]]]];
a[n_] := b[n+1]-b[n];
Table[a[n], {n, 1, 65}] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

Formula

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

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