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 A055664 Norms of Eisenstein-Jacobi primes. 16
 3, 4, 7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 121, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 289, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 529, 541, 547, 571 (list; graph; refs; listen; history; text; internal format)
 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's 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 Consists of 3; rational primes = 1 (mod 3) [A002476]; and squares of rational primes = -1 (mod 3) [A003627^2]. 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 Cf. A055665-A055668, A055025-A055029, A135461, A135462. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice. The Z[sqrt(-5)] analogs are in A020669, A091727, A091728, A091729, A091730 and A091731. Sequence in context: A279815 A088764 A093124 * A089374 A029552 A193883 Adjacent sequences:  A055661 A055662 A055663 * A055665 A055666 A055667 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane, Jun 09 2000 EXTENSIONS More terms from David Wasserman, Mar 21 2002 STATUS approved

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Last modified May 26 19:14 EDT 2019. Contains 323597 sequences. (Running on oeis4.)