A037221 -id:A037221 - cp's OEIS frontend

A037291 Number of rings with 1 containing n elements.

1, 1, 1, 4, 1, 1, 1, 11, 4, 1, 1, 4, 1, 1, 1, 50, 1, 4, 1, 4, 1, 1, 1, 11, 4, 1, 12, 4, 1, 1, 1, 208, 1, 1, 1, 16, 1, 1, 1, 11, 1, 1, 1, 4, 4, 1, 1, 50, 4, 4, 1, 4, 1, 12, 1, 11, 1, 1, 1, 4, 1, 1, 4

Offset: 1

Christian G. Bower, Jun 15 1998

Many authors simply call these "rings". They are also known as unital rings, rings with unity, or rings with identity. - Charles R Greathouse IV, Aug 12 2015

Is this sequence multiplicative? That is, if p and q are distinct primes, is it true that a(p^i*q^j) = a(p^i)*a(q^j)? - Jianing Song, Oct 26 2019. The answer is yes - see the Eric M. Rains link. - N. J. A. Sloane, Oct 27 2019

Gérard P. Michon, Ring Theory, Numericana, 2000-2022.
C. Noebauer, Home page and Table of numbers of small rings [Archived copies]
Eric M. Rains, The number of unital rings with n elements is a multiplicative function of n.

Cf. A027623, A037221, A127707.

a(16) and a(32)-a(63) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000
Keyword 'mult' added by Jianing Song, Feb 02 2020
a(54) corrected by Andrey Zabolotskiy, Feb 02 2023

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

Original entry on oeis.org

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).

A358461 Number of near-rings with identity of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 6, 1, 1, 1, 53, 11, 1, 1, 11, 1, 1, 1, 4274, 1, 26, 1

Offset: 2

Choiwah Chow, Dec 17 2022

a(2) - a(19) were generated by Mace4 with the cube-and-conquer and the invariant algorithms.

G. Pilz, Near-rings: The Theory and Its Applications, Elsevier, 1977.

MathStructures, Near-rings with identity.

Cf. A305858 and A037221 for various sequences related to near-rings.

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A037291 Number of rings with 1 containing n elements.

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

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A358461 Number of near-rings with identity of order n, up to isomorphism.

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