OFFSET
1,1
COMMENTS
These are the norms of the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.
Let us say that an integer n divides a lattice if there exists a sublattice of index n. Example: 3 divides the hexagonal lattice. Then A003136 (Loeschian numbers) is the sequence of divisors of the hexagonal lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. The present sequence gives the prime divisors of the hexagonal lattice. Similarly, A055025 (Norms of Gaussian primes) is the sequence of "prime divisors" of the square lattice. - Jean-Christophe Hervé, Dec 04 2006
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
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
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
Join[{3}, Select[Range[600], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) & ]] (* Jean-François Alcover, Oct 09 2012, from formula *)
PROG
(PARI) is(n)=(isprime(n) && n%3<2) || (issquare(n, &n) && isprime(n) && n%3==2) \\ Charles R Greathouse IV, Apr 30 2013
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
More terms from David Wasserman, Mar 21 2002
STATUS
approved