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A344723 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^5.

%I A344723 #25 May 28 2021 15:57:58
%S A344723 1,31,243,992,3094,7564,16596,31744,58237,97117,158169,241837,364299,
%T A344723 521829,745693,1018120,1389402,1837302,2423834,3105432,3998776,
%U A344723 5007286,6289998,7738784,9543887,11537207,14031231,16717879,20018661,23629281,27958433,32577739,38219963,44148743
%N A344723 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^5.
%C A344723 In general, for m > 1, Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^m ~ (1 - 2^(1-m)) * zeta(m) * n^m. - _Vaclav Kotesovec_, May 28 2021
%H A344723 Seiichi Manyama, <a href="/A344723/b344723.txt">Table of n, a(n) for n = 1..10000</a>
%F A344723 a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^5 - (d - 1)^5).
%F A344723 G.f.: (1/(1 - x)) * Sum_{k>=1} (k^5 - (k - 1)^5) * x^k/(1 + x^k).
%F A344723 a(n) ~ 15*zeta(5)*n^5/16. - _Vaclav Kotesovec_, May 28 2021
%t A344723 a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 27 2021 *)
%t A344723 Accumulate[Table[-3*DivisorSigma[0, n] + 2*DivisorSigma[0, 2*n] + 10*DivisorSigma[1, n] - 5*DivisorSigma[1, 2*n] - 15*DivisorSigma[2, n] + 5*DivisorSigma[2, 2*n] + 25/2 * DivisorSigma[3, n] - 5/2 * DivisorSigma[3, 2*n] - 45/8 *DivisorSigma[4, n] + 5/8 * DivisorSigma[4, 2*n], {n, 1, 50}]] (* _Vaclav Kotesovec_, May 28 2021 *)
%o A344723 (PARI) a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^5);
%o A344723 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^5-(d-1)^5)));
%o A344723 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^5-(k-1)^5)*x^k/(1+x^k))/(1-x))
%Y A344723 Column k=5 of A344726.
%Y A344723 Cf. A318744.
%K A344723 nonn
%O A344723 1,2
%A A344723 _Seiichi Manyama_, May 27 2021