A270998 Table read by rows: list of prime 5-tuples of the form (p, p+2, p+6, p+8, p+12).

5, 7, 11, 13, 17, 11, 13, 17, 19, 23, 101, 103, 107, 109, 113, 1481, 1483, 1487, 1489, 1493, 16061, 16063, 16067, 16069, 16073, 19421, 19423, 19427, 19429, 19433, 21011, 21013, 21017, 21019, 21023, 22271, 22273, 22277, 22279, 22283, 43781, 43783, 43787, 43789, 43793

Offset: 1

Author

Arkadiusz Wesolowski, Jul 12 2016

Keywords

• nonn
• tabf

Comments

A prime 5-tuple is a constellation of five successive primes with distance 12, and is of the form (p, p+2, p+6, p+8, p+12) or (p, p+4, p+6, p+10, p+12).

Initial members p (other than 5) of prime 5-tuples of the form (p, p+2, p+6, p+8, p+12) are congruent to 11 or 101 (mod 210).

Also called prime 5-tuples of the first kind.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• C. K. Caldwell, Top Twenty page, Quintuplet
• Eric Weisstein's World of Mathematics, Prime Constellation
• Wikipedia, Prime quadruplet
• Index entries for primes, gaps between

Crossrefs

Cf. A022006, A270999.

Programs

Magma

lst:=[]; for p in [5..43781 by 2] do if p eq 5 xor p mod 210 in {11, 101} then if IsPrime(p) then t:=[c: c in [p+2..p+12] | IsPrime(c)]; if #t eq 4 then lst:=lst cat [p] cat t; end if; end if; end if; end for; lst;

Mathematica

m = {0, 2, 6, 8, 12}; Union@ Flatten@ Map[# + m &, Select[Prime@ Range[10^4], Times @@ Boole@ PrimeQ[# + m] == 1 &]] (* Michael De Vlieger, Jul 13 2016 *)
Select[Partition[Prime[Range[5000]],5,1],Differences[#]=={2,4,2,4}&]// Flatten (* Harvey P. Dale, Jul 27 2020 *)

Formula

a(5*n-4) = A022006(n).