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

1,2

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.

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).

Example

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

Mathematica

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

Prog

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

Sequence in context: A112967 A375739 A244731 * A109150 A004202 A013647

Adjacent sequences: A307559 A307560 A307561 * A307563 A307564 A307565

Keyword

nonn

Author

Sally Myers Moite, Apr 14 2019

Status

approved