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
Programs
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Magma
[&+[DivisorSigma(2,d)*DivisorSigma(2, n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Oct 16 2019
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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 *) -
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
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
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