cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A342377 Number of rings without 1 containing n elements.

Original entry on oeis.org

0, 1, 1, 7, 1, 3, 1, 41, 7, 3, 1, 18, 1, 3, 3, 340, 1, 18, 1, 18, 3, 3, 1, 93, 7, 3, 47, 18, 1, 7, 1
Offset: 1

Views

Author

Bernard Schott, Mar 12 2021

Keywords

Comments

A ring without 1 is still a ring, although sometimes called a rng, or a non-unital ring, or a pseudo-ring (see Wikipedia links).

Examples

			a(1) = 0 because the only ring with 1 element is the zero ring (see link) with the element 0, and for this ring, 0 and 1 coincide.
a(3) = 1 because the Matrix ring with 3 elements with coefficients from Z/3Z:
         (0 0)       (0 0)        (0 0)
     0 = (0 0),  a = (1 0),   b = (2 0)
  is without 1 (note this ring is commutative) and there is no other such ring with 3 elements and without 1, hence a(3) = 1.

Links

Crossrefs

Number of rings: A037291 (with 1 containing n elements), this sequence (without 1 containing n elements), A027623 or A037234 (with n elements).
Cf. A127707, A342375, A037289, A127708, A342376, A209401.

Formula

a(n) = A037234(n) - A037291(n) = A342375(n) + A342376(n).
a(p) = 1 if p prime.

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.

Original entry on oeis.org

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

Views

Author

José María Grau Ribas, Mar 08 2021

Keywords

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

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}]
Previous Showing 11-12 of 12 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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