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%I #22 Jan 22 2020 06:14:00
%S 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 101,102,121,122,125,137,142,146,165,172,177,181,186,187,205,215,216,
%U 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 Numbers k such that both 6*k + 1 and 6*k + 7 are prime.
%C There are 138 such numbers between 1 and 1000.
%C 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 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 Amiram Eldar, <a href="/A307562/b307562.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 a(3) = 5, so 6(5) + 1 = 31 and 6(5) + 7 = 37 are both prime.
%t Select[Range[400], AllTrue[6 # + {1, 7}, PrimeQ] &] (* _Michael De Vlieger_, Apr 15 2019 *)
%o (PARI) isok(n) = isprime(6*n+1) && isprime(6*n+7); \\ _Michel Marcus_, Apr 16 2019
%Y For the primes see A023201, A046117.
%Y Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
%Y Intersection of A024899 and A153218.
%Y Cf. also A307561, A307563.
%K nonn
%O 1,2
%A _Sally Myers Moite_, Apr 14 2019
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