%I #24 Mar 19 2022 14:52:30

%S 3,4,7,13,19,25,31,37,43,61,67,73,79,97,103,109,121,127,139,151,157,

%T 163,181,193,199,211,223,229,241,271,277,283,289,307,313,331,337,349,

%U 367,373,379,397,409,421,433,439,457,463,487,499,523,529,541,547,571

%N Norms of Eisenstein-Jacobi primes.

%C 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.

%C 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

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

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

%H T. D. Noe, <a href="/A055664/b055664.txt">Table of n, a(n) for n = 1..1000</a>

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

%e 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.

%t Join[{3}, Select[Range[600], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) & ]] (* _Jean-François Alcover_, Oct 09 2012, from formula *)

%o (PARI) is(n)=(isprime(n) && n%3<2) || (issquare(n,&n) && isprime(n) && n%3==2) \\ _Charles R Greathouse IV_, Apr 30 2013

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

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

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_, Jun 09 2000

%E More terms from _David Wasserman_, Mar 21 2002