%I #55 Mar 29 2024 01:36:20
%S 1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,
%T 6227020800,87178291200,1307674368000,24409921536000,355687428096000,
%U 6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000
%N Number of labeled semisimple rings with n elements.
%C Using the Artin-Wedderburn theorem, a finite semisimple ring is a product of matrix algebras over finite field. In particular, if n is squarefree then any semisimple ring of cardinal n is commutative. One can be more precise, indeed all semisimple rings with n elements are commutative if and only if the only 4th power that divides n is 1.
%C The analogous sequences for abelian groups and cyclic groups are A034382 and A034381, respectively.
%C In the case of commutative semisimple rings, we get the factorial numbers.
%H Paul Laubie, <a href="https://github.com/Kellaubz/labeled_semisimple_rings">A Github repository with a code to compute the terms of the form a(p^n)</a>.
%F If n is squarefree then we have a(n) = n!. More precisely, a(n) = n! if and only if the only 4th power that divides n is 1. In particular, n=16 is the smallest n such that a(n) is different from n!.
%F If n and m are relatively prime, then a(n*m) = (n*m)!*a(n)*a(m)/(n!*m!).
%e For n=4, we have two possible rings: F_4 and F_2 X F_2. We use the notation F_q to denote the finite field with q elements. To compute a(4) we need to know how many ring automorphisms F_4 and F_2 X F_2 admit. For F_4, we have that Aut(F_4) is generated by the Frobenius morphism, hence we have 2 automorphisms. For F_2 X F_2, the only nontrivial automorphism is exchanging the two coordinates, hence we also have 2 automorphisms. Hence:
%e a(4) = 24/2 + 24/2 = 24.
%e We can compute a(2^k) for some small values of k:
%e a(4) = 4! = 24,
%e a(8) = 8!,
%e a(16) = 16! + 16!/6,
%e a(32) = 32! + 32!/6,
%e a(64) = 64! + 64!/12 + 64!/12,
%e a(128) = 128! + 128!/36 + 128!/18 + 128!/12,
%e ...
%Y Cf. A000142, A005117, A038538.
%Y Cf. A034382, A034381.
%K nonn
%O 1,2
%A _Paul Laubie_, Mar 05 2024