A345435 - OEIS

%I #68 Dec 16 2021 20:13:52

%S 0,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,23,25,26,28,29,31,32,34,

%T 35,37,39,41,43,45,47,49,51,53,55,57,59,62,63,65,67,68,70,72,73,75,77,

%U 78,80,82,83,85,87,88,90,91,95,97,101,103,107,109,113,115

%N 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.

%C 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

%C 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).

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

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

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

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

%D 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.

%H Rémy Sigrist, <a href="/A345435/b345435.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A345435/a345435.gp.txt">PARI program for A345435</a>

%H N. J. A. Sloane, <a href="/A345435/a345435.pdf">Illustration of initial terms</a> [An enlargement of Figure 1 of Wichmann (2019), showing the numbering of the initial cells of the hexagonal spiral.]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinPrime.html">Eisenstein Prime</a>

%H Brian Wichmann, <a href="http://www.tilingsearch.org/special/ufd.pdf">Tiling for Unique Factorization Domains</a>, Jul 22 2019.

%H Brian Wichmann, <a href="/A345435/a345435.png">The Eisenstein integers, with the primes shaded</a> [Figure 1 from the previous link]

%e 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.

%e The first few Eisenstein primes are (here u is any of the six units):

%e    u*(2+w), norm = 3, number = 6;

%e    2*u, norm = 4, number = 6;

%e    u*(3+w), norm = 7, number = 6;

%e    u*(3+2*w), norm = 7, number = 6 (so there are 12 primes of norm 7 - see A055667).

%o (PARI) See Links section.

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

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jun 23 2021

%E More terms from _Rémy Sigrist_, Jun 26 2021