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

%I A270999 #36 Jun 15 2025 15:37:29
%S A270999 7,11,13,17,19,97,101,103,107,109,1867,1871,1873,1877,1879,3457,3461,
%T A270999 3463,3467,3469,5647,5651,5653,5657,5659,15727,15731,15733,15737,
%U A270999 15739,16057,16061,16063,16067,16069,19417,19421,19423,19427,19429,43777,43781,43783,43787,43789
%N A270999 Table read by rows: list of prime 5-tuples of the form (p, p+4, p+6, p+10, p+12).
%C A270999 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).
%C A270999 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).
%C A270999 Also called prime 5-tuples of the second kind.
%H A270999 Robert Israel, <a href="/A270999/b270999.txt">Table of n, a(n) for n = 1..5665</a>
%H A270999 C. K. Caldwell, Top Twenty page, <a href="https://t5k.org/top20/page.php?id=56">Quintuplet</a>
%H A270999 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeConstellation.html">Prime Constellation</a>
%H A270999 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_quadruplet">Prime quadruplet</a>
%H A270999 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F A270999 a(5*n-4) = A022007(n).
%p A270999 Primes:= select(isprime, [seq(i,i=3..10^5,2)]):
%p A270999 T:= select(t -> Primes[t+4]-Primes[t]=12 and Primes[t+1]-Primes[t]=4, [$1..nops(Primes)-5]):
%p A270999 seq(seq(Primes[t+j],j=0..4),t=T); # _Robert Israel_, Jul 13 2016
%t A270999 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 *)
%t A270999 Select[Partition[Prime[Range[4600]],5,1],Differences[#]=={4,2,4,2}&]//Flatten (* _Harvey P. Dale_, Jun 15 2025 *)
%o A270999 (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;
%o A270999 (MATLAB)
%o A270999 Primes = primes(2*10^8);
%o A270999 T12 = find(Primes(5:end) - Primes(1:end-4)==12);
%o A270999 T4 = find(Primes(2:end) - Primes(1:end-1)==4);
%o A270999 T = intersect(T4,T12);
%o A270999 Primes(reshape([T;T+1;T+2;T+3;T+4],5*numel(T),1)) % _Robert Israel_, Jul 14 2016
%Y A270999 Cf. A022007, A270998.
%K A270999 nonn,tabf
%O A270999 1,1
%A A270999 _Arkadiusz Wesolowski_, Jul 12 2016