A270998 -id:A270998 - cp's OEIS frontend

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

Arkadiusz Wesolowski, Jul 12 2016

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.

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

Cf. A022007, A270998.

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

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;

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

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 *)

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

A275515 Table read by rows: list of prime triples of the form (p, p+2, p+6).

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 41, 43, 47, 101, 103, 107, 107, 109, 113, 191, 193, 197, 227, 229, 233, 311, 313, 317, 347, 349, 353, 461, 463, 467, 641, 643, 647, 821, 823, 827, 857, 859, 863, 881, 883, 887, 1091, 1093, 1097, 1277, 1279, 1283, 1301, 1303, 1307

Offset: 1

Arkadiusz Wesolowski, Jul 31 2016

A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).
Initial members p (other than 5) of prime triples of the form (p, p+2, p+6) are congruent to 11 or 17 (mod 30).
Also called prime triples of the first kind.

			The table starts:
5, 7, 11;
11, 13, 17;
17, 19, 23;
...

C. K. Caldwell, Top Twenty page, Triplet
Eric Weisstein's World of Mathematics, Prime Triplet
Wikipedia, Prime triple
Index entries for primes, gaps between

Cf. A022004, A077800, A136162, A270998, A275516.

&cat[[p, p+2, p+6]: p in PrimesUpTo(1301) | (p le 5 xor p mod 30 in {11, 17}) and IsPrime(p+2) and IsPrime(p+6)];

Prime@ Range[#, # + 2] &@ PrimePi@ Select[Prime@ Range@ 216, Times @@ Boole@ PrimeQ[# + {2, 6}] > 0 &] // Flatten (* Michael De Vlieger, Aug 02 2016 *)

a(3*n-2) = A022004(n).

cp's OEIS Frontend

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

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