A055668 Number of inequivalent Eisenstein-Jacobi primes of norm n.

0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0

Offset

0,8

Comments

These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-omega, +-omega^2).

References

R. K. Guy, Unsolved Problems in Number Theory, A16.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Links

Table of n, a(n) for n=0..104.

Formula

a(n) = 2 if n is a prime = 1 (mod 6); a(n) = 1 if n = 3 or n = p^2 where p is a prime = 2 (mod 3); a(n) = 0 otherwise. - Franklin T. Adams-Watters, May 05 2006

Example

There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

Mathematica

a[3] = 1; a[p_ /; PrimeQ[p] && Mod[p, 6] == 1] = 2; a[n_ /; PrimeQ[p = Sqrt[n]] && Mod[p, 3] == 2] = 1; a[_] = 0; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Aug 19 2013, after Franklin T. Adams-Watters *)
Table[Which[PrimeQ[n]&&Mod[n, 6]==1, 2, n==3, 1, PrimeQ[Sqrt[n]]&&Mod[ Sqrt[ n], 3] == 2, 1, True, 0], {n, 0, 110}] (* Harvey P. Dale, Jun 17 2017 *)

Crossrefs

Cf. A055664-A055667, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

Sequence in context: A369873 A030201 A349797 * A045839 A000086 A363858

Adjacent sequences: A055665 A055666 A055667 * A055669 A055670 A055671

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jun 09 2000

Extensions

More terms from Franklin T. Adams-Watters, May 05 2006

Status

approved