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A344042 a(n) = n * Sum_{d|n} sigma(d)^2 / d.

%I A344042 #18 May 08 2021 08:09:33
%S A344042 1,11,19,71,41,209,71,367,226,451,155,1349,209,781,779,1695,341,2486,
%T A344042 419,2911,1349,1705,599,6973,1166,2299,2278,5041,929,8569,1055,7359,
%U A344042 2945,3751,2911,16046,1481,4609,3971,15047,1805,14839,1979,11005,9266,6589,2351,32205,3746,12826,6479
%N A344042 a(n) = n * Sum_{d|n} sigma(d)^2 / d.
%F A344042 G.f.: Sum_{k >= 1} sigma(k)^2 * x^k/(1 - x^k)^2.
%F A344042 From _Vaclav Kotesovec_, May 08 2021: (Start)
%F A344042 Dirichlet g.f.: zeta(s) * zeta(s-1)^3 * zeta(s-2) / zeta(2*s-2).
%F A344042 Sum_{k=1..n} a(k) ~ 5 * Pi^2 * zeta(3) * n^3 / 36. (End)
%t A344042 a[n_] := n * DivisorSum[n, DivisorSigma[1, #]^2/# &]; Array[a, 51] (* _Amiram Eldar_, May 08 2021 *)
%o A344042 (PARI) a(n) = n*sumdiv(n, d, sigma(d)^2/d);
%o A344042 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^2*x^k/(1-x^k)^2))
%o A344042 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p^2*X^2) / ((1 - X) * (1 - p*X)^3 * (1 - p^2*X)))[n], ", ")) \\ _Vaclav Kotesovec_, May 08 2021
%Y A344042 Cf. A060640, A065018, A226564, A344043.
%K A344042 nonn,mult
%O A344042 1,2
%A A344042 _Seiichi Manyama_, May 08 2021