A319085 a(n) = Sum_{k=1..n} k^2*tau(k), where tau is A000005.

1, 9, 27, 75, 125, 269, 367, 623, 866, 1266, 1508, 2372, 2710, 3494, 4394, 5674, 6252, 8196, 8918, 11318, 13082, 15018, 16076, 20684, 22559, 25263, 28179, 32883, 34565, 41765, 43687, 49831, 54187, 58811, 63711, 75375, 78113, 83889, 89973, 102773, 106135

Offset

1,2

Comments

In general, for m>=1, Sum_{k=1..n} k^m * tau(k) = Sum_{k=1..n} k^m * (Bernoulli(m+1, floor(1 + n/k)) - Bernoulli(m+1, 0)) / (m+1), where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 08 2018

Links

Table of n, a(n) for n=1..41.

Formula

a(n) ~ n^3 * (log(n) + 2*gamma - 1/3)/3, where gamma is the Euler-Mascheroni constant A001620.

a(n) = Sum_{k=1..n} k^2 * Bernoulli(3, floor(1 + n/k)) / 3, where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 08 2018

a(n) = Sum_{k=1..n} Sum_{i=1..floor(n/k)} i^2 * k^2. - Wesley Ivan Hurt, Nov 26 2020

Mathematica

Accumulate[Table[k^2*DivisorSigma[0, k], {k, 1, 50}]]

Prog

(PARI) a(n) = sum(k=1, n, k^2*numdiv(k)); \\ Michel Marcus, Sep 12 2018
(PARI) f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
a(n) = 2*sum(k=1, sqrtint(n), k^2 * f(n\k)) - f(sqrtint(n))^2; \\ Daniel Suteu, Nov 26 2020
(Python)
from math import isqrt
def A319085(n): return (-((s:=isqrt(n))*(s+1)*(2*s+1))**2//12 + sum(k**2*(q:=n//k)*(q+1)*(2*q+1) for k in range(1, s+1)))//3 # Chai Wah Wu, Oct 21 2023

Crossrefs

Cf. A000005, A006218, A034714, A143127.

Sequence in context: A328408 A198956 A110205 * A211531 A264959 A370870

Adjacent sequences: A319082 A319083 A319084 * A319086 A319087 A319088

Keyword

nonn

Author

Vaclav Kotesovec, Sep 10 2018

Status

approved