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

1,1

Links

Table of n, a(n) for n=1..42.

T. Forbes, Prime k-tuplets

Eric Weisstein's World of Mathematics, Prime Triplet.

Formula

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

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

Example

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.

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 *)

Crossrefs

Cf. A078946.

Sequence in context: A307691 A129466 A259516 * A334886 A027399 A266910

Adjacent sequences: A342499 A342500 A342501 * A342503 A342504 A342505

Keyword

nonn

Author

James S. DeArmon, Mar 14 2021

Status

approved