A034761 Dirichlet convolution of sigma(n) with itself.

1, 6, 8, 23, 12, 48, 16, 72, 42, 72, 24, 184, 28, 96, 96, 201, 36, 252, 40, 276, 128, 144, 48, 576, 98, 168, 184, 368, 60, 576, 64, 522, 192, 216, 192, 966, 76, 240, 224, 864, 84, 768, 88, 552, 504, 288, 96, 1608, 178, 588, 288, 644, 108, 1104, 288, 1152, 320, 360

Offset

1,2

Links

T. D. Noe, Table of n, a(n) for n=1..10000

Formula

Dirichlet g.f.: zeta^2(s)*zeta^2(s-1).

Multiplicative with a(2^e) = (e-1) 2^(e+2) + e + 5, a(p^e) = ((1+e)p^(e+3) - (3+e)(p^(e+2)-p+1) + 2)/(p-1)^3, p > 2. - Mitch Harris, Jun 27 2005 [corrected by Amiram Eldar, Oct 16 2022 and Sep 12 2023]

Equals A134577 * A000005. - Gary W. Adamson, Nov 02 2007

Also the Dirichlet convolution A000005 by A038040. - R. J. Mathar, Apr 01 2011

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (2*Pi^2 * log(n) + (4*gamma - 1)*Pi^2 + 24*zeta'(2)) / 144, where gamma is the Euler-Mascheroni constant A001620 and Zeta'(2) = A073002. Equivalently, Sum_{k=1..n} a(k) ~ Pi^4 * n^2 * (2*log(n) - 1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) / 144, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 28 2019

Mathematica

f[p_, e_] := ((e + 1)*p^(e + 3) - (e + 3)*(p^(e + 2) - p + 1) + 2)/(p - 1)^3; f[2, e_] := (e - 1)*2^(e + 2) + e + 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 16 2022 *)

Crossrefs

Cf. A000005, A000203 (sigma), A001620, A073002, A074962, A134577.

Sequence in context: A024868 A262199 A276226 * A085796 A280641 A005887

Adjacent sequences: A034758 A034759 A034760 * A034762 A034763 A034764

Keyword

nonn,mult,changed

Author

Erich Friedman

Status

approved