A305858 - OEIS

A305858

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

1, 3, 5, 35, 10, 99, 24, 3858, 486, 535, 139, 54694, 454, 4537, 3817

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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..15.

Choiwah Chow, Mikoláš Janota, and João Araújo, Cube-based Isomorph-free Finite Model Finding, IOS ebook, Volume 392: ECAI 2024, Frontiers in Artificial Intelligence and Applications. See p. 4105.

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