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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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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
COMMENTS
Define S(k) = Sum_{0<=a<k; 0<=b<k} (a+i*b)^k, where i is the imaginary unit. The sequence contains the places k where S(k) <> 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
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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14, 28, 42, 55, 56, 70, 84, 95, 110, 112, 126, 140, 154, 165, 168, 182, 190, 210, 220, 224, 238, 252, 266, 275, 280, 285, 308, 322, 330, 336, 350, 364, 378, 380, 385, 406, 420, 434, 440, 448, 462, 475, 476, 495, 504, 506, 518, 532, 546, 550, 560, 570, 574, 602, 616, 630, 644, 658, 660, 665, 672, 700, 714, 715, 728, 742
MATHEMATICA
aa[n_] := Mod[Sum[PowerMod[ a + b I, n, n], {a, n}, {b, n}], n]; Select[Range[100], aa[72*#] > 0 &]
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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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
COMMENTS
If m <> 0 (mod 24) then Sum_{(a+b*i)^m: 0<=a<m, 0<=b<m} == 0 (mod m).
MATHEMATICA
aa[n_] := aa[n] = Mod[Sum[PowerMod[a + b *I, n, n], {a, n}, {b, n}], n]; Table[aa[24*n], {n, 1, 10}]
PROG
(PARI) 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
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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1, 0, -144, -2568, -28200, 0, 15203328, 675195936, 16696909080, 0, -25789252433472, -1612260342054360, -54804262577596532, 0, 161017938434267136000, 13718166932451951573120, 621130358284578576358416, 0, -3008072527724272784969384000, -320196271193421334219630013080
MATHEMATICA
g[n_] := Sum[(a + b I)^n, {a, 1, n}, {b, 1, n}]; Table[Re[g[n]], {n, 33}]
PROG
(PARI) vector(100, n, real(sum(x=1, n, sum(y=1, n, (x+I*y)^n)))) \\ Colin Barker, Nov 09 2014
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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1, 18, 144, 0, -28200, -814968, -15203328, 0, 16696909080, 893794451000, 25789252433472, 0, -54804262577596532, -4044941639317807200, -161017938434267136000, 0, 621130358284578576358416, 59512584052525004199214632, 3008072527724272784969384000, 0
MATHEMATICA
g[n_] := Sum[(a + b I)^n, {a, 1, n}, {b, 1, n}]; Table[Im[g[n]], {n, 33}]
PROG
(PARI) vector(100, n, imag(sum(x=1, n, sum(y=1, n, (x+I*y)^n)))) \\ Colin Barker, Nov 09 2014
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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COMMENTS
In most cases S(24*(3*k+1)) == 8*(3*k+1) (mod 24*(3*k+1)).
MATHEMATICA
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) &]
Numbers k such that S(24*(3*k-1)) <> 16*(3*k-1) mod 24*(3*k-1) where S(k) := Sum_{0<=a<n, 0<=b<n} (a+b*i)^n and i is the imaginary unit.
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MATHEMATICA
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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