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 A307562 #22 Jan 22 2020 06:14:00
%S A307562 1,2,5,6,10,11,12,16,17,25,26,32,37,45,46,51,55,61,62,72,76,90,95,100,
%T A307562 101,102,121,122,125,137,142,146,165,172,177,181,186,187,205,215,216,
%U A307562 220,237,241,242,247,257,270,276,277,282,290,291,292,296,297,310,311,312,331,332,335,347,355,356,380,381,390
%N A307562 Numbers k such that both 6*k + 1 and 6*k + 7 are prime.
%C A307562 There are 138 such numbers between 1 and 1000.
%C A307562 Prime pairs that differ by 6 are called "sexy" primes. Other prime pairs that differ by 6 are of the form 6n - 1 and 6n + 5.
%C A307562 Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd - c - d, 6cd + c + d - 1 or 6cd + c + d, that is, they are not (6c - 1)d - c - 1, (6c - 1)d - c, (6c + 1)d + c - 1 or (6c + 1)d + c.
%H A307562 Amiram Eldar, <a href="/A307562/b307562.txt">Table of n, a(n) for n = 1..10000</a>
%H A307562 Sally M. Moite, <a href="http://vixra.org/abs/1903.0353">“Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m</a>, vixra:1903.0353 (2019).
%e A307562 a(3) = 5, so 6(5) + 1 = 31 and 6(5) + 7 = 37 are both prime.
%t A307562 Select[Range[400], AllTrue[6 # + {1, 7}, PrimeQ] &] (* _Michael De Vlieger_, Apr 15 2019 *)
%o A307562 (PARI) isok(n) = isprime(6*n+1) && isprime(6*n+7); \\ _Michel Marcus_, Apr 16 2019
%Y A307562 For the primes see A023201, A046117.
%Y A307562 Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
%Y A307562 Intersection of A024899 and A153218.
%Y A307562 Cf. also A307561, A307563.
%K A307562 nonn
%O A307562 1,2
%A A307562 _Sally Myers Moite_, Apr 14 2019