A319086 a(n) = Sum_{k=1..n} k^2*sigma(k), where sigma is A000203.

1, 13, 49, 161, 311, 743, 1135, 2095, 3148, 4948, 6400, 10432, 12798, 17502, 22902, 30838, 36040, 48676, 55896, 72696, 86808, 104232, 116928, 151488, 170863, 199255, 228415, 272319, 297549, 362349, 393101, 457613, 509885, 572309, 631109, 749045, 801067

Offset: 1

Author

Vaclav Kotesovec, Sep 10 2018

Keywords

• nonn

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

Crossrefs

Cf. A000203, A024916, A143128.

Programs

Mathematica

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

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

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