

A120457


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.


0



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

1,1


COMMENTS

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. AdamsWatters, Aug 20 2011


LINKS

Table of n, a(n) for n=1..45.


FORMULA

a(n) = Sorted[16*Powerof[((Prime[n] + 1)/2)*((Prime[m] + 1)/2)*((Prime[o] + 1)/2)*((Prime[p] + 1)/2)]]


EXAMPLE

q[ 1/2, 1/2, 1/2, 1/2]*q[ 1/2, 1/2, 1/2, 1/2] = {{1,0},{0,1}}


MATHEMATICA

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]]


CROSSREFS

Sequence in context: A064828 A265135 A265136 * A129151 A039546 A223020
Adjacent sequences: A120454 A120455 A120456 * A120458 A120459 A120460


KEYWORD

nonn


AUTHOR

Roger L. Bagula, Jun 24 2006


STATUS

approved



