A344042 a(n) = n * Sum_{d|n} sigma(d)^2 / d.

1, 11, 19, 71, 41, 209, 71, 367, 226, 451, 155, 1349, 209, 781, 779, 1695, 341, 2486, 419, 2911, 1349, 1705, 599, 6973, 1166, 2299, 2278, 5041, 929, 8569, 1055, 7359, 2945, 3751, 2911, 16046, 1481, 4609, 3971, 15047, 1805, 14839, 1979, 11005, 9266, 6589, 2351, 32205, 3746, 12826, 6479

Offset

1,2

Links

Table of n, a(n) for n=1..51.

Formula

G.f.: Sum_{k >= 1} sigma(k)^2 * x^k/(1 - x^k)^2.

From Vaclav Kotesovec, May 08 2021: (Start)

Dirichlet g.f.: zeta(s) * zeta(s-1)^3 * zeta(s-2) / zeta(2*s-2).

Sum_{k=1..n} a(k) ~ 5 * Pi^2 * zeta(3) * n^3 / 36. (End)

Mathematica

a[n_] := n * DivisorSum[n, DivisorSigma[1, #]^2/# &]; Array[a, 51] (* Amiram Eldar, May 08 2021 *)

Prog

(PARI) a(n) = n*sumdiv(n, d, sigma(d)^2/d);
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^2*x^k/(1-x^k)^2))
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p^2*X^2) / ((1 - X) * (1 - p*X)^3 * (1 - p^2*X)))[n], ", ")) \\ Vaclav Kotesovec, May 08 2021

Crossrefs

Cf. A060640, A065018, A226564, A344043.

Sequence in context: A248802 A290078 A089032 * A335410 A106231 A107637

Adjacent sequences: A344039 A344040 A344041 * A344043 A344044 A344045

Keyword

nonn,mult

Author

Seiichi Manyama, May 08 2021

Status

approved