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

1, 31, 243, 992, 3094, 7564, 16596, 31744, 58237, 97117, 158169, 241837, 364299, 521829, 745693, 1018120, 1389402, 1837302, 2423834, 3105432, 3998776, 5007286, 6289998, 7738784, 9543887, 11537207, 14031231, 16717879, 20018661, 23629281, 27958433, 32577739, 38219963, 44148743

Offset: 1

Author

Seiichi Manyama, May 27 2021

Keywords

• nonn

Comments

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Column k=5 of A344726.

Cf. A318744.

Programs

Mathematica

a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
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 *)

PARI

a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^5);

PARI

a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^5-(d-1)^5)));

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))

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^5 - (d - 1)^5).

G.f.: (1/(1 - x)) * Sum_{k>=1} (k^5 - (k - 1)^5) * x^k/(1 + x^k).

a(n) ~ 15*zeta(5)*n^5/16. - Vaclav Kotesovec, May 28 2021