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Search: id:A120457
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| A120457 |
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Sequence of unique powers from a quaternion generalization of Gaussian quadratic reciprocity ( quaternion quartic reciprocity). |
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+0
1
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| 81, 108, 162, 216, 256, 324, 378, 486, 504, 512, 540, 648, 756, 810, 896, 972, 1024, 1080, 1280, 1512, 1620, 1764, 1792, 1944, 2048, 2268, 2520, 2560, 2916, 3136, 3240, 3528, 3584, 3780, 4096, 4480, 4536, 4860, 5120, 5400, 5832, 6272, 6400, 7168, 7560
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Quaternion[ -1/2, 1/2, 1/2, 1/2] is equivalent here to the Gaussian (-1). I've eliminated all the powers that give the identity matrix. These matrices are all unitary (determinant one). When the matrices of these unique powers are sorted they only make 9 types in the first 10^4 products.
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FORMULA
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a(n) = Sorted[16*Powerof[((Prime[n] + 1)/2)*((Prime[m] + 1)/2)*((Prime[o] + 1)/2)*((Prime[p] + 1)/2)]]
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EXAMPLE
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q[ -1/2, 1/2, 1/2, 1/2].q[ -1/2, -1/2, -1/2, -1/2] = {{1,0},{0,1}}
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MATHEMATICA
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i = {{0, 1}, {-1, 0}}; j = {{0, I}, {I, 0}}; k = {{I, 0}, {0, -I}}; e = IdentityMatrix[2]; q[t_, x_, y_, z_] = e*t + x*i + j*y + k*z; f[n_, m_, o_, p_] = ((Prime[n] + 1)/2)*((Prime[m] + 1)/2)*((Prime[o] + 1)/2)*((Prime[p] + 1)/2) a = 16*Union[Flatten[Table[If[MatrixPower[q[ -1/2, 1/2, 1/2, 1/2], f[n, m, o, p]] - e == {{0, 0}, {0, 0}}, {}, f[n, m, o, p]], {n, 1, 10}, {m, 1, 10}, {o, 1, 10}, {p, 1, 10}], 3]]
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CROSSREFS
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Sequence in context: A104113 A102766 A064828 this_sequence A129151 A039546 A053887
Adjacent sequences: A120454 A120455 A120456 this_sequence A120458 A120459 A120460
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KEYWORD
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nonn,uned
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AUTHOR
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Roger Bagula (rlbagulatftn(AT)yahoo.com), Jun 24 2006
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