A055666 Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,3

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

Jean-François Alcover, Table of n, a(n) for n = 1..1000

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

norms = Join[{3}, Select[Range[2000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Length[Reduce[n == a^2 - a*b + b^2, {a, b}, Integers]]/6; A055666 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

Crossrefs

Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

Sequence in context: A282625 A026498 A140685 * A251139 A195061 A064130

Adjacent sequences: A055663 A055664 A055665 * A055667 A055668 A055669

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