A038538 Number of semisimple rings with n elements.

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, 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, 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

Offset: 1

Author

Paolo Dominici (pl.dm(AT)libero.it)

Keywords

• nonn
• nice
• mult

Comments

Enumeration uses Wedderburn-Artin theorem and fact that a finite division ring is a field.

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).

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.
• John Knopfmacher, Abstract analytic number theory, North-Holland, 1975, pp. 63-64.
• T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 2001.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Catalina Calderón and María José Zárate, The Number of Semisimple Rings of Order at most x, Extracta mathematicae, Vol. 7, No. 2-3 (1992), pp. 144-147.
• J. Duttlinger, Eine Bemerkung zu einer asymptotischen Formel von Herrn Knopfmacher, Journal für die reine und angewandte Mathematik, Vol. 1974, No. 266 (1974), pp. 104-106.
• John Knopfmacher, Arithmetical properties of finite rings and algebras, and analytic number theory, Journal für die reine und angewandte Mathematik, Volume 1972, No. 252 (1972), pp. 16-43.
• Werner Georg Nowak, On the value distribution of a class of arithmetic functions, Commentationes Mathematicae Universitatis Carolinae, Vol. 37, No. 1 (1996), pp. 117-134.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A002110, A004101, A005117, A025487, A027623, A052305, A123030 (partial sums), A244285.

Programs

Mathematica

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 *)

PARI

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.
A004101(n) = v004101from1[n];
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A038538(n) = vecproduct(apply(e -> A004101(e), factorint(n)[, 2])); \\ Antti Karttunen, Nov 18 2017

Formula

Multiplicative with a(p^k) = A004101(k).

For all n, a(A002110(n)) = a(A005117(n)) = 1.

From Amiram Eldar, Jan 31 2024: (Start)

Dirichlet g.f.: Product_{k,m>=1} zeta(k*m^2*s).

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)