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%I #45 Apr 18 2023 11:47:19
%S 1,3,13,97,14,39,15,624,67,42,17,1261,18,45,182
%N Number of nonisomorphic rings Z/nZ<x,y>/(x^2 - A, y^2 - B, y*x - a - b*x - c*y - d*x*y) of order n^4.
%H José María Grau Ribas, <a href="/A342305/a342305.pdf">A complete family of non-isomorphic rings of the type Zn(A,B,a,b,c,d)</a>
%H José María Grau Ribas, <a href="/A342305/a342305_3.nb">Mathematica code</a>
%H José María Grau Ribas, Antonio M. Oller-Marcén, and Steve Szabo, <a href="https://doi.org/10.24330/ieja.1281705">Minimal rings related to generalized quaternion rings</a>, Int'l Electronic J. Algebra (2023).
%e For n=2:
%e Z/2Z<x,y>/(x^2, y^2, y*x),
%e Z/2Z<x,y>/(x^2, y^2, y*x + x*y),
%e Z/2Z<x,y>/(x^2, y^2, y*x + 1 + x*y),
%e so a(2)=3.
%e For n=3, a complete family of non-isomorphic cases {A,B,a,b,c,d} are:
%e {0,0,0,0,0,0}, {0,0,0,0,0,1}, {0,0,0,0,0,2}, {0,0,1,0,0,2},
%e {0,1,0,0,0,1}, {0,1,0,0,0,2}, {0,1,0,1,0,0}, {0,2,0,0,0,1}, {0,2,0,0,0,2},
%e {1,0,0,0,1,0}, {1,1,0,0,0,1}, {1,1,1,1,2,0}, {1,2,0,0,0,1},
%e so a(3)=13.
%t a[1]=1; a[p_,1]:= (12 + (p - 1)/2); a[2, 1]=3; a[2,2]= 97; a[2,3]=624; a[3, 2]=67; a[n_]:=Module[{aux=FactorInteger[n]},Product[a[aux[[i,1]], aux[[i,2]]], {i, Length[aux]}]]; Table[a[n], {n, 1, 15}]
%Y Cf. A341548, A341547, A341201, A341202, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948, A000005.
%K nonn,mult,more
%O 1,2
%A _José María Grau Ribas_, Mar 08 2021