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

1,2

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

Cf. A000203, A024916, A143128.

Sequence in context: A009951 A274974 A251142 * A146287 A289999 A147346

Adjacent sequences: A319083 A319084 A319085 * A319087 A319088 A319089

Keyword

nonn

Author

Vaclav Kotesovec, Sep 10 2018

Status

approved