A349770 a(n) = Sum_{d|n} usigma(d) * usigma(n/d).

1, 6, 8, 19, 12, 48, 16, 48, 36, 72, 24, 152, 28, 96, 96, 113, 36, 216, 40, 228, 128, 144, 48, 384, 88, 168, 136, 304, 60, 576, 64, 258, 192, 216, 192, 684, 76, 240, 224, 576, 84, 768, 88, 456, 432, 288, 96, 904, 164, 528, 288, 532, 108, 816, 288, 768, 320, 360, 120, 1824

Offset

1,2

Comments

Dirichlet convolution of A034448 with itself.

Links

Table of n, a(n) for n=1..60.

Formula

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

Multiplicative with a(p^e) = e * (p^e + 1) + (p+1) * (p^e - 1)/(p-1). - Amiram Eldar, Nov 29 2021

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / zeta(3)^2 * (Pi^2 * log(n)/72 + gamma * Pi^2/36 - Pi^2/144 + zeta'(2)/6 - Pi^2 * zeta'(3)/(18*zeta(3))), where zeta(3) = A002117, zeta'(2) = -A073002, zeta'(3) = -A244115 and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2021

Mathematica

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a[n_] := Sum[usigma[d] usigma[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

Crossrefs

Cf. A034448, A034761, A322328, A328485, A343569.

Sequence in context: A108341 A350526 A173975 * A349225 A199884 A028331

Adjacent sequences: A349767 A349768 A349769 * A349771 A349772 A349773

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Nov 29 2021

Status

approved