A328485 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-1).

1, 4, 5, 9, 7, 20, 9, 18, 15, 28, 13, 45, 15, 36, 35, 35, 19, 60, 21, 63, 45, 52, 25, 90, 33, 60, 43, 81, 31, 140, 33, 68, 65, 76, 63, 135, 39, 84, 75, 126, 43, 180, 45, 117, 105, 100, 49, 175, 59, 132, 95, 135, 55, 172, 91, 162, 105, 124, 61, 315, 63, 132, 135, 133, 105

Offset: 1

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

• nonn
• mult

Comments

Inverse Moebius transform of A034448.

Dirichlet convolution of A055615 with A064840.

Links

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

Crossrefs

Cf. A000005, A000203, A007429, A008683, A034448, A048691, A055615, A064840, A319131, A319132.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(d)*tau(n/d)*sigma(n/d)*d, d=divisors(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 16 2019

Mathematica

Table[n DivisorSum[n, MoebiusMu[n/#] DivisorSigma[0, #] DivisorSigma[1, #]/# &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[DivisorSum[k, # &, CoprimeQ[#, k/#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^(e + 1) - p)/(p - 1) + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2023 *)

PARI

a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, (p[i]^(e[i] + 1) - p[i])/(p[i] - 1) + e[i] + 1);} \\ Amiram Eldar, Feb 10 2023

Formula

G.f.: Sum_{k>=1} usigma(k) * x^k / (1 - x^k), where usigma = A034448.

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

a(n) = n * Sum_{d|n} mu(n/d) * tau(d) * sigma(d) / d, where mu = A008683, tau = A000005 and sigma = A000203.

Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (72 * zeta(3)). - Vaclav Kotesovec, Oct 17 2019

From Amiram Eldar, Feb 10 2023: (Start)

a(n) = Sum_{d|n} Sum_{d'|n, gcd(d, d')=1} d'.

Multiplicative with a(p^e) = (p^(e+1)-p)/(p-1) + e + 1. (End)