A307000 -id:A307000 - cp's OEIS frontend

A127707 Number of commutative rings with 1 containing n elements.

1, 1, 1, 4, 1, 1, 1, 10, 4, 1, 1, 4, 1, 1, 1, 37, 1, 4, 1, 4, 1, 1, 1, 10, 4, 1, 11, 4, 1, 1, 1, 109, 1, 1, 1, 16, 1, 1, 1, 10, 1, 1, 1, 4, 4, 1, 1, 37, 4, 4, 1, 4, 1, 11, 1, 10, 1, 1, 1, 4, 1, 1, 4

Offset: 1

Hugues Randriam (randriam(AT)enst.fr), Jan 24 2007

Is this a multiplicative function?

Answer: yes! See the Eric M. Rains link for a proof for the result for all unital rings; restricting to commutative rings does not affect the essence of the proof. - Jianing Song, Feb 02 2020

C. Noebauer, Home page and Table of numbers of small rings [Archived copies from web.archive.org]
Eric M. Rains, The number of unital rings with n elements is a multiplicative function of n.

Cf. A037289, A037291, A127708, A027623, A307000, A341202.

a(n) = A037291(n) - A127708(n). - Bernard Schott, Apr 19 2022

Keyword 'mult' added by Jianing Song, Feb 02 2020
a(32)-a(63) using Nöbauer's data added by Andrey Zabolotskiy, Apr 18 2022
a(32) = 109 corrected by Bernard Schott, Apr 19 2022

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}]

cp's OEIS Frontend

A127707 Number of commutative rings with 1 containing n elements.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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