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%I #22 Jan 31 2024 08:07:23
%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,6,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,8,1,1,
%T 1,4,1,1,1,3,1,1,1,2,2,1,1,6,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,13,1,1,1,2,
%U 1,1,1,6,1,1,2,2,1,1,1,6,6,1,1,2,1,1,1,3,1,2,1,2,1,1,1,8,1,2,2
%N Number of semisimple rings with n elements.
%C Enumeration uses Wedderburn-Artin theorem and fact that a finite division ring is a field.
%C a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.
%D John Knopfmacher, Abstract analytic number theory, North-Holland, 1975, pp. 63-64.
%D T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 2001.
%H Antti Karttunen, <a href="/A038538/b038538.txt">Table of n, a(n) for n = 1..16384</a>
%H Catalina Calderón and María José Zárate, <a href="https://eudml.org/doc/38338">The Number of Semisimple Rings of Order at most x</a>, Extracta mathematicae, Vol. 7, No. 2-3 (1992), pp. 144-147.
%H J. Duttlinger, <a href="https://doi.org/10.1515/crll.1974.266.104">Eine Bemerkung zu einer asymptotischen Formel von Herrn Knopfmacher</a>, Journal für die reine und angewandte Mathematik, Vol. 1974, No. 266 (1974), pp. 104-106.
%H John Knopfmacher, <a href="https://doi.org/10.1515/crll.1972.252.16">Arithmetical properties of finite rings and algebras, and analytic number theory</a>, Journal für die reine und angewandte Mathematik, Volume 1972, No. 252 (1972), pp. 16-43.
%H Werner Georg Nowak, <a href="http://dml.cz/dmlcz/118816">On the value distribution of a class of arithmetic functions</a>, Commentationes Mathematicae Universitatis Carolinae, Vol. 37, No. 1 (1996), pp. 117-134.
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F Multiplicative with a(p^k) = A004101(k).
%F For all n, a(A002110(n)) = a(A005117(n)) = 1.
%F From _Amiram Eldar_, Jan 31 2024: (Start)
%F Dirichlet g.f.: Product_{k,m>=1} zeta(k*m^2*s).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.499616... = A244285 (see A123030 for a more precise asymptotic formula). (End)
%t With[{emax = 7}, f[e_] := f[e] = Coefficient[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[emax]] + 1}, {j, 1, Floor[emax/k^2] + 1}], {x, 0, emax}], x, e]; a[1] = 1; a[n_] := Times @@ f /@ FactorInteger[n][[;; , 2]]; Array[a, 2^emax]] (* _Amiram Eldar_, Jan 31 2024, using code by _Vaclav Kotesovec_ at A004101 *)
%o (PARI)
%o v004101from1 = [1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197]; \\ From the data-section of A004101.
%o A004101(n) = v004101from1[n];
%o vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
%o A038538(n) = vecproduct(apply(e -> A004101(e), factorint(n)[, 2])); \\ _Antti Karttunen_, Nov 18 2017
%Y Cf. A002110, A004101, A005117, A025487, A027623, A052305, A123030 (partial sums), A244285.
%K nonn,nice,mult
%O 1,4
%A Paolo Dominici (pl.dm(AT)libero.it)
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