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A055665
Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).
4
6, 6, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12
OFFSET
1,1
COMMENTS
These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/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
Antti Karttunen, Table of n, a(n) for n = 1..1000 (computed from the b-file of A055666 with the formula of Franklin T. Adams-Watters)
FORMULA
a(n) = 6 * A055666(n) - 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
norms = Join[{3}, Select[Range[1000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Reduce[n == a^2 - a*b + b^2, {a, b}, Integers] // Length; A055665 = 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: A220435 A340512 A338992 * A168328 A176569 A315778
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