A073763 -id:A073763 - cp's OEIS frontend

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

José María Grau Ribas, Oct 15 2013

nonn

Define S(k) = Sum_{0<=a=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

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.

The complement of A230761.

Cf. A230309, A230310, A073763.

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

A230761 Indices k where the sum over k-th powers of the integers in a k X k square in the Gaussian plane (A230308) is not == 0 (mod k).

Original entry on oeis.org

24, 48, 96, 120, 168, 192, 240, 264, 312, 336, 384, 408, 456, 480, 528, 552, 600, 624, 672, 696, 744, 768, 816, 840, 888, 912, 960, 984, 1008, 1032, 1056, 1104, 1128, 1176, 1200, 1248, 1272, 1320, 1344, 1392, 1416, 1464

Offset: 1

Jose Maria Grau Ribas, Oct 16 2013

nonn

Define S(k) = Sum_{0<=a 0 (mod k).
Almost the same as A073763, but contains also 1008 (equivalent to A230310(1)), for example.
The asymptotic density of this sequence is 0.028999... (Fortuny Ayuso et al., 2014). - Amiram Eldar, May 01 2021

Amiram Eldar, Table of n, a(n) for n = 1..300
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.

The complement of A230308.

Cf. A073763, A230309, A230310.

A232820 Imaginary part of the sum over the n-th powers of all Gaussian integers in the n X n base square in the first quadrant.

Original entry on oeis.org

1, 18, 144, 0, -28200, -814968, -15203328, 0, 16696909080, 893794451000, 25789252433472, 0, -54804262577596532, -4044941639317807200, -161017938434267136000, 0, 621130358284578576358416, 59512584052525004199214632, 3008072527724272784969384000, 0

Offset: 1

José María Grau Ribas, Nov 30 2013

sign

Colin Barker, Table of n, a(n) for n = 1..200

Cf. A230308-A230310, A073763, A232819.

g[n_] := Sum[(a + b I)^n, {a, 1, n}, {b, 1, n}]; Table[Im[g[n]], {n, 33}]

vector(100, n, imag(sum(x=1, n, sum(y=1, n, (x+I*y)^n)))) \\ Colin Barker, Nov 09 2014

Conjecture: a(4n) = 0. - Michel Marcus, Nov 09 2014

cp's OEIS Frontend

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).

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A230761 Indices k where the sum over k-th powers of the integers in a k X k square in the Gaussian plane (A230308) is not == 0 (mod k).

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A232820 Imaginary part of the sum over the n-th powers of all Gaussian integers in the n X n base square in the first quadrant.

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