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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).
8
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
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.
Sequence in context: A135382 A351831 A328617 * A357875 A064598 A366187
KEYWORD
nonn
AUTHOR
STATUS
approved