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A344722 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^4.

%I A344722 #22 May 28 2021 17:17:59
%S A344722 1,15,81,240,610,1230,2336,3840,6371,9455,14097,19615,27441,36205,
%T A344722 48849,61874,79860,99470,124816,150846,186498,221646,267232,313840,
%U A344722 373059,431599,508595,581009,673635,767835,881357,989615,1131667,1264111,1429875,1590464,1785010
%N A344722 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^4.
%H A344722 Seiichi Manyama, <a href="/A344722/b344722.txt">Table of n, a(n) for n = 1..10000</a>
%F A344722 a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^4 - (d - 1)^4).
%F A344722 G.f.: (1/(1 - x)) * Sum_{k>=1} (k^4 - (k - 1)^4) * x^k/(1 + x^k).
%F A344722 a(n) ~ 7 * Pi^4 * n^4 / 720. - _Vaclav Kotesovec_, May 28 2021
%t A344722 a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 27 2021 *)
%t A344722 Accumulate[Table[3*DivisorSigma[0, n] - 2*DivisorSigma[0, 2*n] - 8*DivisorSigma[1, n] + 4*DivisorSigma[1, 2*n] + 9*DivisorSigma[2, n] - 3*DivisorSigma[2, 2*n] - 5*DivisorSigma[3, n] + DivisorSigma[3, 2*n], {n, 1, 50}]] (* _Vaclav Kotesovec_, May 28 2021 *)
%o A344722 (PARI) a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^4);
%o A344722 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^4-(d-1)^4)));
%o A344722 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^4-(k-1)^4)*x^k/(1+x^k))/(1-x))
%Y A344722 Column k=4 of A344726.
%Y A344722 Cf. A318743.
%K A344722 nonn
%O A344722 1,2
%A A344722 _Seiichi Manyama_, May 27 2021