A270999 Table read by rows: list of prime 5-tuples of the form (p, p+4, p+6, p+10, p+12).

7, 11, 13, 17, 19, 97, 101, 103, 107, 109, 1867, 1871, 1873, 1877, 1879, 3457, 3461, 3463, 3467, 3469, 5647, 5651, 5653, 5657, 5659, 15727, 15731, 15733, 15737, 15739, 16057, 16061, 16063, 16067, 16069, 19417, 19421, 19423, 19427, 19429, 43777, 43781, 43783, 43787, 43789

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 7) of prime 5-tuples of the form (p, p+4, p+6, p+10, p+12) are congruent to 97 or 187 (mod 210).

Also called prime 5-tuples of the second kind.

Links

• Robert Israel, Table of n, a(n) for n = 1..5665
• 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. A022007, A270998.

Programs

MATLAB

Primes = primes(2*10^8);
T12 = find(Primes(5:end) - Primes(1:end-4)==12);
T4 = find(Primes(2:end) - Primes(1:end-1)==4);
T = intersect(T4,T12);
Primes(reshape([T;T+1;T+2;T+3;T+4],5*numel(T),1)) % Robert Israel, Jul 14 2016

Magma

lst:=[]; for p in [5..43777 by 2] do if p le 7 xor p mod 210 in {97, 187} then if IsPrime(p) then t:=[c: c in [p+4..p+12] | IsPrime(c)]; if #t eq 4 then lst:=lst cat [p] cat t; end if; end if; end if; end for; lst;

Maple

Primes:= select(isprime, [seq(i,i=3..10^5,2)]):
T:= select(t -> Primes[t+4]-Primes[t]=12 and Primes[t+1]-Primes[t]=4, [$1..nops(Primes)-5]):
seq(seq(Primes[t+j],j=0..4),t=T); # Robert Israel, Jul 13 2016

Mathematica

m = {0, 4, 6, 10, 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[4600]],5,1],Differences[#]=={4,2,4,2}&]//Flatten (* Harvey P. Dale, Jun 15 2025 *)

Formula

a(5*n-4) = A022007(n).