cp's OEIS Frontend

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.

%I A345436 #20 Aug 06 2021 08:46:52
%S A345436 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,
%T A345436 51,53,59,61,67,69,75,77,81,83,87,89,91,93,97,99,101,103,107,109,111,
%U A345436 113,117,119,121,125,127,131,133,137,139,143,145,149,151,155,157
%N 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.
%C A345436 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
%C A345436 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).
%C A345436 The norms of the Gaussian primes are listed in A055025, and the number of primes with a given norm is given in A055026.
%C A345436 The successive norms of the Gaussian integers along the square spiral are listed in A336336.
%D A345436 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.2, p. 106.
%D A345436 L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
%D A345436 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 A345436 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian prime</a>.
%H A345436 Brian Wichmann, <a href="http://www.tilingsearch.org/special/ufd.pdf">Tiling for Unique Factorization Domains</a>, Jul 22 2019. See Fig. 2.
%Y A345436 Equals A308412 - 1. Cf. A345435, A345437.
%Y A345436 Cf. also A003173, A004018, A055025, A055026, A103431, A174344, A274923, A336336.
%K A345436 nonn
%O A345436 1,2
%A A345436 _N. J. A. Sloane_, Jun 23 2021
%E A345436 Name clarified by _Peter Munn_, Aug 02 2021