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A339948 Number of non-isomorphic generalized quaternion rings over Z/nZ.

%I A339948 #39 Jan 21 2021 10:03:59
%S A339948 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,
%T A339948 20,40,4,64,4,52,16,16,16
%N A339948 Number of non-isomorphic generalized quaternion rings over Z/nZ.
%C A339948 Generalized quaternion rings over Z/nZ are of the form Z_n<x,y>/(x^2-a, y^2-b, xy+yx).
%H A339948 José María Grau Ribas, <a href="/A339948/a339948.pdf">Generalized quaternion rings over Z/nZ for an odd n</a>
%H A339948 Jose María Grau, C. Miguel and A. M. Oller-Marcen, <a href="https://arxiv.org/abs/1706.04760">Generalized Quaternion Rings over Z/nZ for an odd n</a>, arXiv:1706.04760 [math.RA], 2017.
%H A339948 J. M. Grau, C. Miguel and A. M. Oller-Marcén, <a href="https://doi.org/10.1007/s00006-018-0839-x">Generalized quaternion rings over Z/nZ for an odd n</a>, Advances in Applied Clifford Algebras, 28(1), (2018) article 17.
%F A339948 If n is odd then a(n) = A286779(n).
%e A339948 For n=2 all such rings are isomorphic to Z_n<x,y>/(x^2, y^2, xy+yx), so a(2)=1.
%e A339948 For n=4:
%e A339948   Z_n<x,y>/(x^2,   y^2,   xy+yx),
%e A339948   Z_n<x,y>/(x^2,   y^2-1, xy+yx),
%e A339948   Z_n<x,y>/(x^2,   y^2-2, xy+yx),
%e A339948   Z_n<x,y>/(x^2,   y^2-3, xy+yx),
%e A339948   Z_n<x,y>/(x^2-1, y^2-1, xy+yx),
%e A339948   Z_n<x,y>/(x^2-1, y^2-2, xy+yx),
%e A339948   Z_n<x,y>/(x^2-3, y^2-3, xy+yx),
%e A339948 so a(4)=7.
%t A339948 Clear[phi]; phi[1] = phi[2] = 1; phi[4] = 7; phi[8] = 16;
%t A339948 phi[16] = 36; phi[p_, s_] := 2 s^2 + 2;
%t A339948 phi[n_] :=  Module[{aux = FactorInteger[n]},Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];
%t A339948 Table[phi[i], {i,1, 35}]
%Y A339948 Cf. A286779, A027623, A037221, A127708, A037291, A127707, A038538, A037289, A037290, A038036.
%K A339948 nonn,mult,hard,more
%O A339948 1,3
%A A339948 _José María Grau Ribas_, Dec 24 2020