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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 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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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
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
MATHEMATICA
aa[n_] := aa[n] = Mod[Sum[PowerMod[a + b I, n, n], {a, n}, {b, n}], n]; Select[Range[100], aa[#] == 0 &]
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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7
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 &]
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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2
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 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.
+10
1
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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