A034718 Dirichlet convolution of b_n=n with b_n with b_n.

1, 6, 9, 24, 15, 54, 21, 80, 54, 90, 33, 216, 39, 126, 135, 240, 51, 324, 57, 360, 189, 198, 69, 720, 150, 234, 270, 504, 87, 810, 93, 672, 297, 306, 315, 1296, 111, 342, 351, 1200, 123, 1134, 129, 792, 810, 414, 141, 2160, 294, 900, 459, 936, 159, 1620, 495

Offset: 1

Author

Erich Friedman

Keywords

• nonn
• mult
• easy

Comments

Row sums of triangle A329323. - Omar E. Pol, Nov 21 2019

Links

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

Crossrefs

Cf. A000027, A001620, A007425, A038040, A082633, A329323.

Programs

Mathematica

Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 31 2018 *)
f[p_, e_] := (e+1)*(e+2)*p^e/2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 29 2020 *)

Formula

a(n) = Sum_{k*l*m = n} k*l*m, for positive integers k, l, m. This equals one sixth of the same sum over all integers. - Ralf Stephan, May 06 2005

Dirichlet g.f.: zeta^3(x-1).

Multiplicative with a(p^e) = p^e * binomial(e+2, 2). - Mitch Harris, Jun 27 2005

a(n) = n*A007425(n). Dirichlet convolution of A000027 by A038040. - R. J. Mathar, Mar 30 2011

Sum_{k=1..n} a(k) ~ (2*log(n)^2 + (12*gamma - 2)*log(n) + 12*gamma^2 - 6*gamma - 12*sg1 + 1) * n^2 / 8, where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Sep 11 2019

G.f.: Sum_{k>=1} k*tau(k)*x^k / (1 - x^k)^2, where tau = A000005. - Ilya Gutkovskiy, Sep 22 2020