%I #28 Aug 31 2021 02:44:13
%S 0,0,0,2,0,0,0,17,2,0,0,4,0,0,0,215,0,4,0,4,0,0,0,35,2,0,23,4,0,0,0
%N Number of non-commutative rings without 1 containing n elements.
%C 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).
%C These are rings in which multiplication has no unit, and where there is at least one pair of non-commuting elements.
%C a(n)=0 if and only if n is squarefree.
%H Gregory Dresden, <a href="https://web.archive.org/web/20170310135758/http://home.wlu.edu/~dresdeng/smallrings/matrices.html">Rings of Size Four</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pseudo-ring">Pseudo-ring</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rng_(algebra)">Rng</a>.
%H <a href="/index/Res#rings">Index to sequences related to rings</a>.
%F a(n) = A209401(n) - A127708(n) = A342377(n) - A342375(n).
%F a(A005117(n)) = 0; a(A013929(n)) > 0.
%e For n=4, there are 11 rings of order 4, 2 of which are without 1 and non-commutative, so a(4)= 2. Note that for these 2 rings, the abelian group under addition is the commutative Klein group Z/2Z + Z/2Z. These two rings are the last two rings described in the link _Greg Dresden_ in reference: Ring 22.NC.1 and Ring 22.NC.2.
%Y Number of non-commutative rings: A127708 (with 1 containing n elements), this sequence (without 1 containing n elements), A209401 (with n elements).
%Y Cf. A127707, A342375, A037289, A037291, A342377, A027623, A037234.
%Y Cf. A005117, A013929.
%K nonn,more
%O 1,4
%A _Bernard Schott_, Mar 10 2021
%E a(28) corrected by _Des MacHale_, Mar 20 2021