A364269 a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.

1, 41, 311, 1655, 4905, 15705, 32855, 76375, 142714, 272714, 435096, 797976, 1171466, 1857466, 2734966, 4131702, 5556472, 8210032, 10692990, 15060990, 19691490, 26186770, 32635280, 44385680, 54557555, 69497155, 85637215, 108686815, 129222353, 164322353

Offset: 1

Author

Seiichi Manyama, Oct 20 2023

Keywords

• nonn
• easy

Links

• Wikipedia, Faulhaber's formula.

Crossrefs

Cf. A356125, A364268.

Cf. A000537, A001157.

Programs

Mathematica

Accumulate[Table[n^3*DivisorSigma[2, n], {n, 1, 30}]] (* Amiram Eldar, Oct 20 2023 *)

PARI

f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=3, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));

Python

def A364269(n): return sum(k*(k**2*(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 A364269(n): return ((((s:=isqrt(n))*(s+1))**4*(1-s*(s+1<<1))>>2) + sum(((q:=n//k)*(q+1))**2*k**3*(3*k**2+(q*(q+1<<1)-1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^5 * A000537(floor(n/k)).

a(n) ~ (zeta(3)/6) * n^6. - Amiram Eldar, Oct 20 2023