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a(n) = Sum_{k=1..n} sigma_2(k) * floor(n/k).
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%I #15 Aug 07 2022 04:07:52

%S 1,7,18,45,72,138,189,301,403,565,688,985,1156,1462,1759,2212,2503,

%T 3115,3478,4207,4768,5506,6037,7269,7947,8973,9895,11272,12115,13897,

%U 14860,16678,18031,19777,21154,23908,25279,27457,29338,32362,34045,37411,39262,42583

%N a(n) = Sum_{k=1..n} sigma_2(k) * floor(n/k).

%F a(n) = Sum_{k=1..n} Sum_{d|k} d^2 * tau(k/d).

%F G.f.: (1/(1-x)) * Sum_{k>=1} sigma_2(k) * x^k/(1 - x^k).

%F a(n) ~ zeta(3)^2 * n^3 / 3. - _Vaclav Kotesovec_, Aug 07 2022

%t Table[Sum[DivisorSigma[2, k]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* _Vaclav Kotesovec_, Aug 07 2022 *)

%o (PARI) a(n) = sum(k=1, n, sigma(k, 2)*(n\k));

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, d^2*numdiv(k/d)));

%o (PARI) my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, 2)*x^k/(1-x^k))/(1-x))

%Y Partial sums of A007433.

%Y Column k=2 of A356045.

%Y Cf. A000005 (tau).

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jul 24 2022