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A344721 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^3.

%I A344721 #22 May 28 2021 17:17:52
%S A344721 1,7,27,56,118,196,324,448,685,901,1233,1549,2019,2445,3157,3664,4482,
%T A344721 5262,6290,7128,8536,9598,11118,12392,14255,15743,18087,19711,22149,
%U A344721 24417,27209,29251,32771,35327,39087,42048,46046,49244,54180,57512,62434,66838,72258,76246
%N A344721 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^3.
%H A344721 Seiichi Manyama, <a href="/A344721/b344721.txt">Table of n, a(n) for n = 1..10000</a>
%F A344721 a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^3 - (d - 1)^3).
%F A344721 G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 + x^k).
%F A344721 a(n) ~ 3*zeta(3)*n^3/4. - _Vaclav Kotesovec_, May 28 2021
%t A344721 a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 27 2021 *)
%t A344721 Accumulate[Table[-3*DivisorSigma[0, n] + 2*DivisorSigma[0, 2*n] + 6*DivisorSigma[1, n] - 3*DivisorSigma[1, 2*n] - 9/2 * DivisorSigma[2, n] + 3/2 * DivisorSigma[2, 2*n], {n, 1, 50}]] (* _Vaclav Kotesovec_, May 28 2021 *)
%o A344721 (PARI) a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^3);
%o A344721 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^3-(d-1)^3)));
%o A344721 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1+x^k))/(1-x))
%Y A344721 Column k=3 of A344726.
%Y A344721 Cf. A318742.
%K A344721 nonn
%O A344721 1,2
%A A344721 _Seiichi Manyama_, May 27 2021