A345435 Represent the ring of Eisenstein integers E = {x+y*omega: x, y rational integers, omega = exp(2*Pi*i/3)} by the cells of a hexagonal grid; number the cells of the grid along a counterclockwise hexagonal spiral, with the cells 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

0, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 91, 95, 97, 101, 103, 107, 109, 113, 115

Offset

1,2

Comments

The Eisenstein integer represented by cell m is A307013(m) + A307012(m)*omega. Thus the set of Eisenstein primes is {A307013(a(n)) + A307012(a(n))*omega : n >= 2}. - Peter Munn, Jun 26 2021

The Eisenstein integer a + b*omega has norm a^2 - a*b + b^2 (see A003136). The number of Eisenstein integers of norm n is given by A004016(n).

The norms of the Eisenstein primes are given in A055664, and the number of Eisenstein primes of norm n is given in A055667.

Reid's 1910 book (still in print) is still the best reference for the Eisenstein integers and similar rings.

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.4, p. 111.

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

H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A345435

N. J. A. Sloane, Illustration of initial terms [An enlargement of Figure 1 of Wichmann (2019), showing the numbering of the initial cells of the hexagonal spiral.]

Eric Weisstein's World of Mathematics, Eisenstein Prime

Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019.

Brian Wichmann, The Eisenstein integers, with the primes shaded [Figure 1 from the previous link]

Example

The smallest Eisenstein integers are 0 (of norm 0), and the six units of norm 1, namely (writing w for omega) +-1, +-w, +-w^2.
The first few Eisenstein primes are (here u is any of the six units):
   u*(2+w), norm = 3, number = 6;
   2*u, norm = 4, number = 6;
   u*(3+w), norm = 7, number = 6;
   u*(3+2*w), norm = 7, number = 6 (so there are 12 primes of norm 7 - see A055667).

Prog

(PARI) See Links section.

Crossrefs

Cf. A003136, A003173, A003627, A004016, A007645, A055664, A055667, A307012, A307013, A308412, A345436, A345437.

Sequence in context: A363783 A168097 A020719 * A162787 A182308 A023388

Adjacent sequences: A345432 A345433 A345434 * A345436 A345437 A345438

Keyword

nonn

Author

N. J. A. Sloane, Jun 23 2021

Extensions

More terms from Rémy Sigrist, Jun 26 2021

Status

approved