A305858 a(n) = number of near-rings with n elements.

1, 3, 5, 35, 10, 99, 24, 3858, 486, 535, 139, 54694, 454

Offset

1,2

Comments

Near-rings are defined like rings but addition need not be commutative and multiplication need only left-distribute over addition (of course, right-distribution leads to an equivalent theory). Also, there need not exist a multiplicative identity.

References

Gunter Pilz, Near-Rings: The Theory and its Applications, Revised Edition (1983), North-Holland Publishing Company.

Links

Table of n, a(n) for n=1..13.

Choiwah Chow, Mikoláš Janota, and João Araújo, Cube-based Isomorph-free Finite Model Finding, Czech Tech. Univ. (Czechia, 2024). See p. 6.

Formula

Since all rings are near-rings, a(n) >= A027623(n).

Example

The only near-ring of order 1 is the trivial ring, so a(1) = 1.
There are 3 near-rings of order 2, all over Z2, so a(2) = 3.
There are 5 near-rings of order 3, all over Z3, so a(3) = 5.
There are 12 near-rings over Z4 and 23 near-rings over Z2^2, so a(4) = 12 + 23 = 35.
There are 10 near-rings of order 5, all over Z5, so a(5) = 10.
There are 60 near-rings over Z6 and 39 near-rings over S3, so a(6) = 60 + 39 = 99.
There are 24 near-rings of order 7, all over Z7, so a(7) = 24.
There are 135 near-rings over Z8, 1447 near-rings over D8, 281 near-rings over Q, 115 over Z4*Z2, and many over Z2^3, so a(8) > 1978.

Crossrefs

Cf. A027623, A000001.

Sequence in context: A221158 A068111 A162444 * A261659 A346715 A259853

Adjacent sequences: A305855 A305856 A305857 * A305859 A305860 A305861

Keyword

nonn,hard,more

Author

Charles R Greathouse IV, Jun 10 2018

Extensions

a(8)-a(13) from Choiwah Chow, Dec 18 2022

Status

approved