A342305 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.

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

Offset

1,2

Links

Table of n, a(n) for n=1..15.

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).

Example

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.

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}]

Crossrefs

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

Sequence in context: A183283 A006898 A277492 * A275528 A129375 A293528

Adjacent sequences: A342302 A342303 A342304 * A342306 A342307 A342308

Keyword

nonn,mult,more

Author

José María Grau Ribas, Mar 08 2021

Status

approved