OFFSET
1,2
LINKS
Wikipedia, Faulhaber's formula.
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
MATHEMATICA
Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Oct 20 2023
STATUS
approved