A230308 -id:A230308 - cp's OEIS frontend

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).

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.

A230309 Sum_{0<=a<24n, 0<=b<24n} (a+bi)^(24n) (mod 24*n), where i is the imaginary unit.

Original entry on oeis.org

8, 32, 0, 32, 80, 0, 56, 128, 0, 80, 176, 0, 104, 272, 0, 128, 272, 0, 152, 320, 0, 176, 368, 0, 200, 416, 0, 416, 464, 0, 248, 512, 0, 272, 560, 0, 296, 608, 0, 320, 656, 432, 344, 704, 0, 368, 752, 0, 392, 800, 0, 416, 848, 0, 560, 320, 0, 464, 944, 0, 488

Offset: 1

José María Grau Ribas, Oct 16 2013

nonn

If m <> 0 (mod 24) then Sum_{(a+b*i)^m: 0<=a

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A230308, A230310.

aa[n_] := aa[n] = Mod[Sum[PowerMod[a + b *I, n, n], {a, n}, {b, n}], n]; Table[aa[24*n],{n,1,10}]

a(n)=my(N=24*n,a);lift(sum(A=0,N-1,a=Mod(A,N);sum(b=0,N-1,(a+b*I)^N))) \\ Charles R Greathouse IV, Nov 05 2013

A232819 Real 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, 0, -144, -2568, -28200, 0, 15203328, 675195936, 16696909080, 0, -25789252433472, -1612260342054360, -54804262577596532, 0, 161017938434267136000, 13718166932451951573120, 621130358284578576358416, 0, -3008072527724272784969384000, -320196271193421334219630013080

Offset: 1

José María Grau Ribas, Nov 30 2013

sign

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

Cf. A230308-A230310, A232820.

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

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

a(A016825(n)) = 0, for n>=0. - Michel Marcus, Nov 09 2014

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

A232056 Numbers k such that S(24(3k+1)) !== 8(3k+1) (mod 24(3k+1)) where S(j) := Sum_{a=0..j-1, b=0..j-1} (a+bi)^j and i is the imaginary unit; i.e., A230309(3k+1) != 8(3k+1).

Original entry on oeis.org

9, 18, 23, 37, 51

Offset: 1

José María Grau Ribas, Nov 17 2013

In most cases S(24*(3*k+1)) == 8*(3*k+1) (mod 24*(3*k+1)).

Cf. A230308, A230309, A230310, A232057.

fu[n_] := fu[n] = Mod[Sum[PowerMod[i + j I, n, n], {i, 0, n - 1}, {j, 0, n - 1}], n]; Select[Range[50], ! fu[24*(3 # +1)] == 8*(3 # +1) &]

A232057 Numbers k such that S(24(3k-1)) <> 16(3k-1) mod 24(3k-1) where S(k) := Sum_{0<=a

Original entry on oeis.org

5, 19, 37, 47, 61, 75

Offset: 1

José María Grau Ribas, Nov 17 2013

Cf. A230308, A230309, A230310, A232056.

fu[n_] := fu[n] = Mod[Sum[PowerMod[i + j I, n, n], {i, 0, n - 1}, {j, 0, n - 1}], n]; Table[If[! fu[24(3n - 1)]/(3n - 1) == 16, Print[n]; n], {n, 1, 40}]

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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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A230310 Numbers k such that Sum_{a=0..72*k-1, b=0..72*k-1} (a+b*i)^(72*k) !== 0 (mod 72*k), where i is the imaginary unit.

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A230309 Sum_{0<=a<24*n, 0<=b<24*n} (a+b*i)^(24*n) (mod 24*n), where i is the imaginary unit.

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A232819 Real 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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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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A232056 Numbers k such that S(24*(3*k+1)) !== 8*(3*k+1) (mod 24*(3*k+1)) where S(j) := Sum_{a=0..j-1, b=0..j-1} (a+b*i)^j and i is the imaginary unit; i.e., A230309(3*k+1) != 8*(3*k+1).

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A232057 Numbers k such that S(24*(3*k-1)) <> 16*(3*k-1) mod 24*(3*k-1) where S(k) := Sum_{0<=a

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A230310 Numbers k such that Sum_{a=0..72k-1, b=0..72k-1} (a+bi)^(72k) !== 0 (mod 72*k), where i is the imaginary unit.

A230309 Sum_{0<=a<24n, 0<=b<24n} (a+bi)^(24n) (mod 24*n), where i is the imaginary unit.

A232056 Numbers k such that S(24(3k+1)) !== 8(3k+1) (mod 24(3k+1)) where S(j) := Sum_{a=0..j-1, b=0..j-1} (a+bi)^j and i is the imaginary unit; i.e., A230309(3k+1) != 8(3k+1).

A232057 Numbers k such that S(24(3k-1)) <> 16(3k-1) mod 24(3k-1) where S(k) := Sum_{0<=a