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A120457
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Sorted list of numbers of the form (p+1)*(q+1)*(r+1)*(s+1) that are not multiples of 48, where p, q, r, and s are primes.
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0
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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
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OFFSET
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1,1
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COMMENTS
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Original name: Sequence of unique powers from a quaternion generalization of Gaussian quadratic reciprocity (quaternion quartic reciprocity).
The 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. [This comment needs to be clarified]
This quaternion is a primitive cube root of 1; its powers behave like any other primitive cube root of 1. We are thus looking at products (p+1)*(q+1)*(r+1)*(s+1) modulo 48, where p, q, r, and s are primes. The only possible odd result is with all 4 primes = 2: 81. If no prime = 2, the result must be a multiple of 16, which gives three residues, one of which is the identity. If 2 and another prime are present, the result is a multiple of 6, which produces 8 residues; again, one is the identity. - Franklin T. Adams-Watters, Aug 20 2011
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LINKS
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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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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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