A356042 a(n) = Sum_{k=1..n} sigma_2(k) * floor(n/k).

1, 7, 18, 45, 72, 138, 189, 301, 403, 565, 688, 985, 1156, 1462, 1759, 2212, 2503, 3115, 3478, 4207, 4768, 5506, 6037, 7269, 7947, 8973, 9895, 11272, 12115, 13897, 14860, 16678, 18031, 19777, 21154, 23908, 25279, 27457, 29338, 32362, 34045, 37411, 39262, 42583

Offset

1,2

Links

Table of n, a(n) for n=1..44.

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} d^2 * tau(k/d).

G.f.: (1/(1-x)) * Sum_{k>=1} sigma_2(k) * x^k/(1 - x^k).

a(n) ~ zeta(3)^2 * n^3 / 3. - Vaclav Kotesovec, Aug 07 2022

Mathematica

Table[Sum[DivisorSigma[2, k]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 07 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, sigma(k, 2)*(n\k));
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, d^2*numdiv(k/d)));
(PARI) my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, 2)*x^k/(1-x^k))/(1-x))
(Python)
from math import isqrt
def A356042(n):
    c, j, v = 0, 1, 0
    while j <= n:
        k = n//j
        m = n//k
        c += k*(-v+(v:=-(s:=isqrt(m))**2*(s+1)*(2*s+1)+sum((q:=m//a)*(6*a**2+q*(2*q+3)+1) for a in range(1, s+1))))
        j = m+1
    return c//6 # Chai Wah Wu, May 21 2026

Crossrefs

Partial sums of A007433.

Column k=2 of A356045.

Cf. A000005 (tau).

Sequence in context: A023166 A002764 A124053 * A324900 A343545 A324944

Adjacent sequences: A356039 A356040 A356041 * A356043 A356044 A356045

Keyword

nonn,changed

Author

Seiichi Manyama, Jul 24 2022

Status

approved