A308412 -id:A308412 - cp's OEIS frontend

A345435 Represent the ring of Eisenstein integers E = {x+yomega: x, y rational integers, omega = exp(2Pi*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

N. J. A. Sloane, Jun 23 2021

nonn

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.

			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).

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.

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]

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

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More terms from Rémy Sigrist, Jun 26 2021

A345436 Represent the ring of Gaussian integers E = {x+y*i: x, y rational integers, i = sqrt(-1)} by the cells of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells representing the ring identities 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Original entry on oeis.org

0, 2, 4, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 59, 61, 67, 69, 75, 77, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157

Offset: 1

N. J. A. Sloane, Jun 23 2021

nonn

The cell with spiral index m represents the Gaussian integer A174344(m+1) + A274923(m+1) * i. So the set of Gaussian primes is {A174344(a(n)+1) + A274923(a(n)+1) * i : n >= 2}. - Peter Munn, Aug 02 2021
The Gaussian integer z = x+i*y has norm x^2+y^2. There are four units (of norm 1), +-1, +-i. The number of Gaussian integers of norm n is A004018(n).
The norms of the Gaussian primes are listed in A055025, and the number of primes with a given norm is given in A055026.
The successive norms of the Gaussian integers along the square spiral are listed in A336336.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.2, p. 106.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
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.

Eric Weisstein's World of Mathematics, Gaussian prime.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Fig. 2.

Equals A308412 - 1. Cf. A345435, A345437.

Cf. also A003173, A004018, A055025, A055026, A103431, A174344, A274923, A336336.

Name clarified by Peter Munn, Aug 02 2021

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