A342305 Number of nonisomorphic rings Z/nZ/(x^2 - A, y^2 - B, y*x - a - b*x - c*y - d*x*y) of order n^4.

1, 3, 13, 97, 14, 39, 15, 624, 67, 42, 17, 1261, 18, 45, 182

Offset: 1

Author

José María Grau Ribas, Mar 08 2021

Keywords

• nonn
• mult
• more

Examples

			For n=2:
  Z/2Z<x,y>/(x^2, y^2, y*x),
  Z/2Z<x,y>/(x^2, y^2, y*x + x*y),
  Z/2Z<x,y>/(x^2, y^2, y*x + 1 + x*y),
so a(2)=3.
For n=3, a complete family of non-isomorphic cases {A,B,a,b,c,d} are:
  {0,0,0,0,0,0}, {0,0,0,0,0,1}, {0,0,0,0,0,2}, {0,0,1,0,0,2},
  {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},
  {1,0,0,0,1,0}, {1,1,0,0,0,1}, {1,1,1,1,2,0}, {1,2,0,0,0,1},
so a(3)=13.
		

Links

• José María Grau Ribas, A complete family of non-isomorphic rings of the type Zn(A,B,a,b,c,d)
• José María Grau Ribas, Mathematica code
• José María Grau Ribas, Antonio M. Oller-Marcén, and Steve Szabo, Minimal rings related to generalized quaternion rings, Int'l Electronic J. Algebra (2023).

Crossrefs

Cf. A341548, A341547, A341201, A341202, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948, A000005.

Programs

Mathematica

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}]