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A230308 Numbers k such that the sum over the k-th powers of all Gaussian integers in the k X k base square in the first quadrant is == 0 (mod k).

%I A230308 #28 May 01 2021 06:54:14
%S A230308 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,
%T A230308 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,
%U A230308 52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70
%N A230308 Numbers k such that the sum over the k-th powers of all Gaussian integers in the k X k base square in the first quadrant is == 0 (mod k).
%C A230308 Define S(k) = Sum_{0<=a<k, 0<=b<k} (a+b*i)^k, where i is the imaginary unit, which yields S(k) mod k = 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8,.. for k>=1. Then this sequence contains all places k such that S(k) == 0 (mod k).
%C A230308 The asymptotic density of this sequence is 0.971000... (Fortuny Ayuso et al., 2014). - _Amiram Eldar_, May 01 2021
%H A230308 Amiram Eldar, <a href="/A230308/b230308.txt">Table of n, a(n) for n = 1..10000</a>
%H A230308 Pedro Fortuny Ayuso, Jose Maria Grau and Antonio Oller-Marcen, <a href="http://arxiv.org/abs/1402.0333">A von Staudt-type formula for Sum_{z in Zn[i]} z^k</a>, arXiv:1402.0333 [math.NT], 2014.
%t A230308 aa[n_] := aa[n] = Mod[Sum[PowerMod[a + b I, n, n], {a,n}, {b, n}], n];Select[Range[100], aa[#] == 0 &]
%Y A230308 The complement of A230761.
%Y A230308 Cf. A230309, A230310, A073763.
%K A230308 nonn
%O A230308 1,1
%A A230308 _José María Grau Ribas_, Oct 15 2013