A305858 - OEIS

%I #26 Oct 26 2024 14:18:32

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

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

%C 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.

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

%H Choiwah Chow, Mikoláš Janota, and João Araújo, <a href="https://doi.org/10.3233/FAIA240980">Cube-based Isomorph-free Finite Model Finding</a>, IOS ebook, Volume 392: ECAI 2024, Frontiers in Artificial Intelligence and Applications. See p. 4105.

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

%e The only near-ring of order 1 is the trivial ring, so a(1) = 1.

%e There are 3 near-rings of order 2, all over Z2, so a(2) = 3.

%e There are 5 near-rings of order 3, all over Z3, so a(3) = 5.

%e There are 12 near-rings over Z4 and 23 near-rings over Z2^2, so a(4) = 12 + 23 = 35.

%e There are 10 near-rings of order 5, all over Z5, so a(5) = 10.

%e There are 60 near-rings over Z6 and 39 near-rings over S3, so a(6) = 60 + 39 = 99.

%e There are 24 near-rings of order 7, all over Z7, so a(7) = 24.

%e 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.

%Y Cf. A027623, A000001.

%K nonn,hard,more,changed

%O 1,2

%A _Charles R Greathouse IV_, Jun 10 2018

%E a(8)-a(13) from _Choiwah Chow_, Dec 18 2022

%E a(14)-a(15) from _Choiwah Chow_, Oct 21 2024