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A339948
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Number of non-isomorphic generalized quaternion rings over Z/nZ.
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6
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1, 1, 4, 7, 4, 16, 4, 16, 10, 16, 4, 40, 4, 16, 16, 36, 4, 40, 4, 40, 16, 16, 4, 80, 10, 16, 20, 40, 4, 64, 4, 52, 16, 16, 16
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OFFSET
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1,3
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COMMENTS
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Generalized quaternion rings over Z/nZ are of the form Z_n<x,y>/(x^2-a, y^2-b, xy+yx).
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LINKS
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FORMULA
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If n is odd then a(n) = A286779(n).
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EXAMPLE
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For n=2 all such rings are isomorphic to Z_n<x,y>/(x^2, y^2, xy+yx), so a(2)=1.
For n=4:
Z_n<x,y>/(x^2, y^2, xy+yx),
Z_n<x,y>/(x^2, y^2-1, xy+yx),
Z_n<x,y>/(x^2, y^2-2, xy+yx),
Z_n<x,y>/(x^2, y^2-3, xy+yx),
Z_n<x,y>/(x^2-1, y^2-1, xy+yx),
Z_n<x,y>/(x^2-1, y^2-2, xy+yx),
Z_n<x,y>/(x^2-3, y^2-3, xy+yx),
so a(4)=7.
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MATHEMATICA
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Clear[phi]; phi[1] = phi[2] = 1; phi[4] = 7; phi[8] = 16;
phi[16] = 36; phi[p_, s_] := 2 s^2 + 2;
phi[n_] := Module[{aux = FactorInteger[n]}, Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];
Table[phi[i], {i, 1, 35}]
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CROSSREFS
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KEYWORD
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nonn,mult,hard,more
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AUTHOR
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STATUS
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approved
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