A038036 -id:A038036 - cp's OEIS frontend

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

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, 20, 40, 4, 64, 4, 52, 16, 16, 16

Offset: 1

José María Grau Ribas, Dec 24 2020

Generalized quaternion rings over Z/nZ are of the form Z_n/(x^2-a, y^2-b, xy+yx).

			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.

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.

Cf. A286779, A027623, A037221, A127708, A037291, A127707, A038538, A037289, A037290, A038036.

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}]

If n is odd then a(n) = A286779(n).

A341201 Number of unitary rings with additive group (Z/nZ)^3.

Original entry on oeis.org

1, 7, 7, 27, 7, 49, 7

Offset: 1

José María Grau Ribas, Feb 07 2021

R. Gilmer and J. Mott, Associative rings of order_P^3, Proc. Jpn. Acad., 49, No. 10, (1973), 725-799.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

Cf. A341202, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948.

A341202 Number of unitary commutative rings with additive group (Z/nZ)^3.

Original entry on oeis.org

1, 6, 6, 16, 6, 36, 6

Offset: 1

José María Grau Ribas, Feb 07 2021

R. Gilmer and J. Mott, Associative rings of order P^3, Proc. Jpn. Acad., 49, No. 10, (1973), 725-799.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

Cf. A341201, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948.

A341547 Number of rings with additive group (Z/nZ)^2.

Original entry on oeis.org

1, 8, 8, 66, 8, 64, 8, 301, 175, 64, 8, 528, 8, 64, 64

Offset: 1

José María Grau Ribas, Feb 14 2021

B. Fine, Classification of Finite Rings of Order p^2, Mathematics Magazine, 66(4) (1993), 248-252.
C. R. Fletcher, Rings of Small Order, The Mathematical Gazette, 64(427) (1980), 9-22.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

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

Clear[phi]; phi[1] = 1; phi[p_,1] := 8; phi[2,2] = 66;
phi[2,3] = 301; phi[3,2] = 175; phi[n_]:= Module[{aux = FactorInteger[n]}, Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];

A341548 Number of commutative rings with additive group (Z/nZ)^2.

Original entry on oeis.org

1, 6, 6, 28, 6, 36, 6, 79, 35, 36, 6, 168, 6, 36, 36

Offset: 1

José María Grau Ribas, Feb 14 2021

It appears that a(16)=230, but it is preferable to wait for someone to confirm it.

B. Fine, Classification of Finite Rings of Order p^2, Mathematics Magazine, 66(4) (1993), 248-252.
C. R. Fletcher, Rings of Small Order, The Mathematical Gazette, 64(427) (1980), 9-22.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

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

Clear[phi]; phi[p_, 1] := 6;  phi[2,2] = 28; phi[2,3] = 79;  phi[3,2] = 35; phi[n_]:= Module[{aux = FactorInteger[n]}, Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];

A342305 Number of nonisomorphic rings Z/nZ/(x^2 - A, y^2 - B, yx - a - bx - cy - dx*y) of order n^4.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Mar 08 2021

			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.

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).

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

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}]

A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 72, 12151, 53146457

Offset: 0

Jianing Song, Oct 26 2019

For the number of group-like algebraic structures of order n, see:
Semigroups: A027851 or A001423 (commutative: A001426);
Monoids: A058129 or A058133 (commutative: A058131);
Quasigroups: A057991 or A058171 (commutative: A057992);
Loops: A057771 or this sequence (commutative: A089925);
Groups: A000001 (commutative: A000688);
Rings: A027623 or A038036 (commutative: A037289);
Rings with unity: A037291;
Fields: A069513.

a(n) = (A057771(n)+A057996(n))/2.

cp's OEIS Frontend

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

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A341201 Number of unitary rings with additive group (Z/nZ)^3.

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A341202 Number of unitary commutative rings with additive group (Z/nZ)^3.

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A341547 Number of rings with additive group (Z/nZ)^2.

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A341548 Number of commutative rings with additive group (Z/nZ)^2.

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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.

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A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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Author

Keywords

Crossrefs

Formula

A342305 Number of nonisomorphic rings Z/nZ/(x^2 - A, y^2 - B, yx - a - bx - cy - dx*y) of order n^4.