OFFSET
1,3
COMMENTS
Generalized quaternion rings over Z/nZ are of the form Z_n<x,y>/(x^2-a, y^2-b, xy+yx).
LINKS
José María Grau Ribas, Generalized quaternion rings over Z/nZ for an odd n
Jose María Grau, C. Miguel and A. M. Oller-Marcen, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017.
J. M. Grau, C. Miguel and A. M. Oller-Marcén, Generalized quaternion rings over Z/nZ for an odd n, Advances in Applied Clifford Algebras, 28(1), (2018) article 17.
FORMULA
If n is odd then a(n) = A286779(n).
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn,mult,hard,more
AUTHOR
José María Grau Ribas, Dec 24 2020
STATUS
approved