A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.

1, 25, 133, 581, 1331, 3923, 6667, 14347, 23824, 41824, 57796, 106180, 136938, 202794, 283794, 410770, 499204, 726652, 863832, 1199832, 1496184, 1879512, 2171520, 3000960, 3485335, 4223527, 5010847, 6240159, 6971829, 8915829, 9869141, 11933525, 13658501

Offset: 1

Author

Seiichi Manyama, Oct 20 2023

Keywords

• nonn
• easy

Links

• Wikipedia, Faulhaber's formula.

Crossrefs

Partial sums of A282211.

Cf. A319086, A320895, A356125, A364269.

Cf. A000203, A000537, A143128.

Programs

Mathematica

Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* 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=1) = sum(k=1, n, k^(s+t)*f(n\k, s));

Python

def A364194(n): return sum((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 A364194(n): return ((((s:=isqrt(n))*(s + 1))**3*(2*s+1)*(1-3*s*(s+1))>>1) + sum((q:=n//k)*(q+1)*k**3*(q*(15*k+q*(15*k+12*q+18)+2)-2) for k in range(1,s+1)))//60 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^4 * A000537(floor(n/k)).

a(n) ~ (zeta(2)/5) * n^5. - Amiram Eldar, Oct 20 2023