A123030 -id:A123030 - cp's OEIS frontend

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

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

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).

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.

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.

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

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 *)

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

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)

A244285 Decimal expansion of A1B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{rm^2 > 1} zeta(r*m^2).

Original entry on oeis.org

2, 4, 9, 9, 6, 1, 6, 1, 1, 2, 9, 3, 6, 2, 9, 8, 2, 7, 4, 9, 3, 2, 3, 7, 3, 8, 2, 1, 7, 5, 6, 4, 9, 8, 0, 3, 4, 5, 7, 0, 4, 0, 2, 2, 5, 8, 8, 0, 7, 5, 9, 5, 4, 4, 3, 2, 0, 6, 2, 1, 0, 9, 4, 8, 1, 2, 1, 2, 2, 4, 3, 6, 8, 1, 6, 9, 6, 5, 1, 3, 6, 4, 7, 2, 6, 8, 8, 6, 3, 3, 6, 4, 3, 0, 9, 7, 5, 3, 6, 2, 8, 7, 2, 2, 6

Offset: 1

Jean-François Alcover, Jun 25 2014

The asymptotic mean of A038538. - Amiram Eldar, Jan 31 2024

			2.499616112936298274932373821756498034570402258807595443206210948121224368...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.

Cf. A021002 (A1), A038538, A123030.

Digits := 200: z:=product(Zeta(1.0*j), j = 2..1000): for k from 10 by 10 to 50 do print(z*product(product(Zeta(1.0*r*m^2), r = 1..k^2), m = 2..k)); end do; # Vaclav Kotesovec, Jun 11 2020

digits = 20; digitsPlus = 100; n0 = 50; dn = 1; A1 = NProduct[Zeta[m], {m, 2, Infinity}, WorkingPrecision -> digitsPlus]; Clear[B1]; B1[n_] := B1[n] = NProduct[Zeta[r*m^2], {r, 1, n}, {m, 2, n}, WorkingPrecision -> digitsPlus]; B1[n0]; B1[n = n0 + dn]; While[ RealDigits[B1[n], 10, digitsPlus] != RealDigits[B1[n - dn], 10, digitsPlus], Print["n = ", n]; n = n + dn]; RealDigits[A1*B1[n], 10, digits] // First

prodinf(m = 2, zeta(m)) * prodinf(r = 1, prodinf(m = 2, zeta(r*m^2))) \\ Amiram Eldar, Jan 31 2024

More digits from Vaclav Kotesovec, Jun 11 2020

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A038538 Number of semisimple rings with n elements.

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A244285 Decimal expansion of A1B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{rm^2 > 1} zeta(r*m^2).

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cp's OEIS Frontend

A038538 Number of semisimple rings with n elements.

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Formula

A244285 Decimal expansion of A1*B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{r*m^2 > 1} zeta(r*m^2).

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Maple

Mathematica

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Extensions

A244285 Decimal expansion of A1B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{rm^2 > 1} zeta(r*m^2).