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).

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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset

1,1

Comments

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).

The asymptotic density of this sequence is 0.971000... (Fortuny Ayuso et al., 2014). - Amiram Eldar, May 01 2021

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Pedro Fortuny Ayuso, Jose Maria Grau and Antonio Oller-Marcen, A von Staudt-type formula for Sum_{z in Zn[i]} z^k, arXiv:1402.0333 [math.NT], 2014.

Mathematica

aa[n_] := aa[n] = Mod[Sum[PowerMod[a + b I, n, n], {a, n}, {b, n}], n]; Select[Range[100], aa[#] == 0 &]

Crossrefs

The complement of A230761.

Cf. A230309, A230310, A073763.

Sequence in context: A135382 A351831 A328617 * A357875 A064598 A366187

Adjacent sequences: A230305 A230306 A230307 * A230309 A230310 A230311

Keyword

nonn

Author

José María Grau Ribas, Oct 15 2013

Status

approved