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A053778 First of four consecutive primes that comprise two sets of twin primes.

%I A053778 #23 Feb 16 2025 08:32:42
%S A053778 5,11,101,137,179,191,419,809,821,1019,1049,1481,1871,1931,2081,2111,
%T A053778 2969,3251,3359,3371,3461,4217,4229,4259,5009,5651,5867,6689,6761,
%U A053778 6779,6947,7331,7547,8219,8969,9419,9431,9437,10007,11057,11159,11699,12239
%N A053778 First of four consecutive primes that comprise two sets of twin primes.
%C A053778 These twins are not necessarily at the minimal distance as in A007530 (which is a subsequence).
%H A053778 M. F. Hasler, <a href="/A053778/b053778.txt">Table of n, a(n) for n = 1..5274</a>.
%H A053778 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadruplet.html">Prime Quadruplet</a>
%F A053778 A001359 primes for which A048614 is zero. Lesser of 2-twin primes after which the consecutive prime difference pattern (of A001223) is [2, 6k-2, 2] for some k.
%e A053778 These primes initiate consecutive p quadruples as follows: [p,p+2,p+6k,p+6k+2]. For 6k=6,12,18,24,30,36,54 such a p =5,137,1931,9437,2968, 20441 and 48677 resp. Such a quadruple is [48677,48679,48731,48733], with [2,52,2] difference pattern.
%t A053778 Transpose[Select[Partition[Prime[Range[1500]],4,1],#[[4]]-#[[3]]== #[[2]]-#[[1]]== 2&]][[1]] (* _Harvey P. Dale_, Jul 07 2011 *)
%o A053778 (PARI) forprime( p=1,10^5, isprime(p+2) || next; isprime(nextprime(p+4)+2) && print1(p","))
%o A053778 (PARI) nextA053778(p)=until( isprime(nextprime(p+1)+2), until( p+2==p=nextprime(p+1),)); p-2
%o A053778 (PARI) p=0; A053778=vector(100,i, p=nextA053778(p+1))
%Y A053778 Cf. A001223, A001359, A007530, A048614.
%K A053778 nonn
%O A053778 1,1
%A A053778 _Labos Elemer_, Mar 24 2000
%E A053778 Edited by _N. J. A. Sloane_, Apr 13 2008, at the suggestion of M. F. Hasler.