A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

1, 10, 20, 67, 52, 200, 100, 380, 282, 520, 244, 1340, 340, 1000, 1040, 1973, 580, 2820, 724, 3484, 2000, 2440, 1060, 7600, 1978, 3400, 3460, 6700, 1684, 10400, 1924, 9710, 4880, 5800, 5200, 18894, 2740, 7240, 6800, 19760, 3364, 20000, 3700, 16348, 14664

Offset

1,2

Comments

Dirichlet convolution of A001157 with itself.

Dirichlet convolution of A000005 with A034714.

Dirichlet convolution of A000290 with A007433.

Links

Table of n, a(n) for n=1..45.

Formula

a(n) = Sum_{d|n} sigma_2(d) * sigma_2(n/d), where sigma_2 = A001157.

a(n) = Sum_{d|n} d^2 * tau(d) * tau(n/d), where tau = A000005.

Sum_{k=1..n} a(k) ~ zeta(3) * n^3 * (zeta(3)*(log(n)/3 + 2*gamma/3 - 1/9) + 2*zeta'(3)/3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019

Multiplicative with a(p^e) = ((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3. - Amiram Eldar, Sep 15 2023

Mathematica

Table[DivisorSum[n, DivisorSigma[2, #] DivisorSigma[2, n/#] &], {n, 1, 45}]
f[p_, e_] :=((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3 ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

Prog

(Magma) [&+[DivisorSigma(2, d)*DivisorSigma(2, n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Oct 16 2019
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 / (1 - p^2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

Crossrefs

Cf. A000005, A000290, A001157, A001620, A007426, A007433, A034714, A034761.

Cf. A175705.

Sequence in context: A174183 A267554 A337434 * A131726 A276764 A215878

Adjacent sequences: A328487 A328488 A328489 * A328491 A328492 A328493

Keyword

nonn,easy,mult

Author

Ilya Gutkovskiy, Oct 16 2019

Status

approved