A055664 Norms of Eisenstein-Jacobi primes.

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

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

TheGrayCuber, Eisenstein Primes Visually, Youtube video.

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.

Cf. A102283 ({kronecker(-3,n)}), whose inverse Moebius transform A002324 gives the numbers of distinct ideals (or non-associate elements) with each norm (i.e., the coefficients of Dedekind zeta function).

Cf. A007645 (primes not inert in Q(sqrt(-3))), A002476 (primes decomposing), A003627 (primes remaining inert), A045309 (primes not decomposing).

Norms of prime ideals in the ring of integers of quadratic fields of class number 1: A391371 (D=24), A391370 (D=21), A391369 (D=12), A055673 (D=8), A341783 (D=5), this sequence (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341787 (D=-19), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163).

Norms of prime ideals in the ring of integers of quadratic fields of class number 2: A391367 (D=40), A341786 (D=-15), A091727 (D=-20), A391366 (D=-24).

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