A307562 Numbers k such that both 6*k + 1 and 6*k + 7 are prime.

1, 2, 5, 6, 10, 11, 12, 16, 17, 25, 26, 32, 37, 45, 46, 51, 55, 61, 62, 72, 76, 90, 95, 100, 101, 102, 121, 122, 125, 137, 142, 146, 165, 172, 177, 181, 186, 187, 205, 215, 216, 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

Offset: 1

Author

Sally Myers Moite, Apr 14 2019

Keywords

• nonn

Comments

There are 138 such numbers between 1 and 1000.

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.

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.

Examples

			a(3) = 5, so 6(5) + 1 = 31 and 6(5) + 7 = 37 are both prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Sally M. Moite, “Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

Crossrefs

For the primes see A023201, A046117.

Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.

Intersection of A024899 and A153218.

Cf. also A307561, A307563.

Programs

Mathematica

Select[Range[400], AllTrue[6 # + {1, 7}, PrimeQ] &] (* Michael De Vlieger, Apr 15 2019 *)

PARI

isok(n) = isprime(6*n+1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019