A342377 Number of rings without 1 containing n elements.

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

1,4

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).

Links

Table of n, a(n) for n=1..31.

Wikipedia, Pseudo-ring.

Wikipedia, Rng.

Wikipedia, Zero ring.

Index to sequences related to rings.

Formula

a(n) = A037234(n) - A037291(n) = A342375(n) + A342376(n).

a(p) = 1 if p prime.

Example

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.

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.

Sequence in context: A317834 A340144 A340141 * A346103 A039616 A382687

Adjacent sequences: A342374 A342375 A342376 * A342378 A342379 A342380

Keyword

nonn,more

Author

Bernard Schott, Mar 12 2021

Status

approved