A364869 Numbers k such that 6*k+1 is the norm of an Eisenstein prime.

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 20, 21, 23, 25, 26, 27, 30, 32, 33, 35, 37, 38, 40, 45, 46, 47, 48, 51, 52, 55, 56, 58, 61, 62, 63, 66, 68, 70, 72, 73, 76, 77, 81, 83, 87, 88, 90, 91, 95, 96, 100, 101, 102, 103, 105, 107, 110, 112, 115, 118, 121, 122, 123

Offset

1,2

Comments

Numbers k such that 6*k+1 is a prime or the square of a prime congruent to 5 modulo 6.

If p is an Eisenstein prime of norm 6*a(n)+1 (there are two up to association if a(n) is a prime, one if a(n) is the square of a prime), then for any Eisenstein integer x, we have x^a(n) == 0, 1, w, w^2, -1, -w or -w^2 (mod p), where w = (1+sqrt(-3))/2 is a primitive sixth root of unity.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Formula

a(n) = (A055664(n+2) - 1)/6.

Example

4 is a term since 6*4+1 is the norm of the Eisenstein prime 5.

Prog

(PARI) isA364869(n) = isA055664(6*n+1) \\ See A055664 for its program

Crossrefs

Cf. A055664, A364868.

Contains 4*A024702 as a subsequence.

Sequence in context: A175857 A173919 A205666 * A245809 A064295 A001751

Adjacent sequences: A364866 A364867 A364868 * A364870 A364871 A364872

Keyword

nonn,easy

Author

Jianing Song, Aug 11 2023

Status

approved