This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A052378 #27 May 09 2024 13:26:15
%S A052378 7,13,37,97,103,223,307,457,853,877,1087,1297,1423,1483,1867,1993,
%T A052378 2683,3457,4513,4783,5227,5647,6823,7873,8287,10453,13687,13873,15727,
%U A052378 16057,16063,16183,17383,19417,19423,20743,21013,21313,22273,23053,23557
%N A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.
%C A052378 The sequence includes A052166, A052168, A022008 and also other primes like 13, 103, 16063 etc.
%C A052378 a(n) is the lesser term of a 4-twin (A023200) after which the next 4-twin comes in minimal distance [here it is 2; see A052380(4/2)].
%C A052378 Analogous prime sequences are A047948, A052376, A052377 and A052188-A052198 with various [d, A052380(d/2), d] difference patterns following a(n).
%C A052378 All terms == 1 (mod 6) - _Zak Seidov_, Aug 27 2012
%C A052378 Subsequence of A022005. - _R. J. Mathar_, May 06 2017
%H A052378 Zak Seidov, <a href="/A052378/b052378.txt">Table of n, a(n) for n = 1..2000</a>
%F A052378 a(n) is the initial prime of a [p, p+4, p+6, p+6+4] prime-quadruple consisting of two 4-twins: [p, p+4] and [p+6, p+10].
%e A052378 103 initiates [103,107,109,113] prime quadruple followed by [4,2,4] difference pattern.
%t A052378 a = {}; Do[If[Prime[x + 3] - Prime[x] == 10, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* _Zerinvary Lajos_, Apr 03 2007 *)
%t A052378 Select[Partition[Prime[Range[3000]],4,1],Differences[#]=={4,2,4}&][[All,1]] (* _Harvey P. Dale_, Jun 16 2017 *)
%o A052378 (PARI) is(n)=n%6==1 && isprime(n+4) && isprime(n+6) && isprime(n+10) && isprime(n) \\ _Charles R Greathouse IV_, Apr 29 2015
%Y A052378 Cf. A023200, A053320, A022008, A052166, A052168, A001223, A052380.
%K A052378 nonn
%O A052378 1,1
%A A052378 _Labos Elemer_, Mar 22 2000