A244285 -id:A244285 - 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)

A123030 Partial sums of A038538.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 26, 27, 29, 30, 32, 33, 34, 35, 38, 40, 41, 44, 46, 47, 48, 49, 57, 58, 59, 60, 64, 65, 66, 67, 70, 71, 72, 73, 75, 77, 78, 79, 85, 87, 89, 90, 92, 93, 96, 97, 100, 101, 102, 103, 105, 106, 107, 109, 122, 123, 124, 125, 127, 128

Offset: 1

Jonathan Vos Post, Jul 07 2008

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A038538, A244285 (A_1*B_1).

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]]; Accumulate@ Array[a, 2^emax]] (* Amiram Eldar, Jan 31 2024, using code by Vaclav Kotesovec at A004101 *)

a(n) = A_1*B_1*n + A_2*B_2*n^(1/2) + A_3*B_3*n^(1/3) + O(n^(50/199+eps)), where A_k = Product_{m>=1, m!=k} zeta(m/k) and B_k = Product_{r>=1, m>=2} zeta(r*m^2/k) (Finch, 2003). - Amiram Eldar, Jan 31 2024

A335494 Decimal expansion of s, where s is the root of Product_{k>=1} zeta(k*s) = 2.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 11 2020

			1.8868691498777028184721160405750829854817368935543730821633756671766...

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Cf. A021002, A107311, A244285, A329385.

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A038538 Number of semisimple rings with n elements.

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A123030 Partial sums of A038538.

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A335494 Decimal expansion of s, where s is the root of Product_{k>=1} zeta(k*s) = 2.

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