OFFSET
1,2
COMMENTS
In general, for m>=1, Sum_{k=1..n} k^m * sigma(k) = Sum_{k=1..n} k^(m+1) * (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
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) ~ Pi^2 * n^4/24.
a(n) = Sum_{k=1..n} ((k/2) * floor(n/k) * floor(1 + n/k))^2. - Daniel Suteu, Nov 07 2018
MATHEMATICA
Accumulate[Table[k^2*DivisorSigma[1, k], {k, 1, 50}]]
PROG
(PARI) a(n) = sum(k=1, n, k^2*sigma(k)); \\ Michel Marcus, Sep 12 2018
(Python)
def A319086(n): return sum((k*(m:=n//k)*(m+1)>>1)**2 for k in range(1, n+1)) # Chai Wah Wu, Oct 20 2023
(Python)
from math import isqrt
def A319086(n): return ((-((s:=isqrt(n))*(s+1))**3*(2*s+1)>>1) + sum(k**2*(q:=n//k)*(q+1)*(2*k*(2*q+1)+3*q*(q+1)) for k in range(1, s+1)))//12 # Chai Wah Wu, Oct 21 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Sep 10 2018
STATUS
approved