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A364194
a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.
1
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
1,2
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
Partial sums of A282211.
Sequence in context: A232792 A305956 A367104 * A317217 A147489 A156158
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Oct 20 2023
STATUS
approved