A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

1, 2, 4, 5, 9, 10, 12, 15, 17, 22, 25, 29, 32, 39, 44, 45, 60, 65, 67, 72, 75, 80, 82, 94, 95, 99, 100, 109, 114, 117, 120, 124, 127, 137, 152, 155, 164, 169, 172, 177, 185, 194, 199, 204, 205, 214, 215, 220, 229, 240, 242, 247, 254, 260, 262, 267, 269, 270, 289, 304, 312, 330, 334, 347, 355, 359, 369, 374, 379, 389

Offset: 1

Author

Sally Myers Moite, Apr 14 2019

Keywords

• nonn

Comments

There are 140 such numbers between 1 and 1000.

These numbers correspond to all the prime pairs which differ by 8 except 3 and 11.

Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd + c - d, 6cd - c + d or 6cd + c + d - 1, that is, they are not (6c - 1)d - c - 1, (6c - 1)d + c, (6c + 1)d - c or (6c + 1)d + c - 1.

Examples

			a(4) = 5, so 6(5) - 1 = 29 and 6(5) + 7 = 37 are both prime.
		

Links

• Robert Israel, 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

The primes are A023202, A092402, A031926.

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

Intersection of A024898 and A153218.

Cf. also A307561, A307562.

Programs

Maple

select(t -> isprime(6*t-1) and isprime(6*t+7), [$1..500]); # Robert Israel, May 27 2019

PARI

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