1,1

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 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 HERVE (jcherve(AT)ifn.fr), Dec 04 2006

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.

T. D. Noe, Table of n, a(n) for n=1..1000

Consists of 3; rational primes = 1 (mod 3) [A002476]; and squares of rational primes = -1 (mod 3) [A003627^2 ]

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.

Cf. A055665-A055668, A055025-A055029, A135461, A135462. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

The Z[sqrt(-5)] analogues are in A020669, A091727, A091728, A091729, A091730 and A091731.

Sequence in context: A076784 A088764 A093124 this_sequence A089374 A029552 A125118

Adjacent sequences: A055661 A055662 A055663 this_sequence A055665 A055666 A055667

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Jun 09 2000

More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Mar 21 2002