OFFSET
1,2
COMMENTS
Conjecture: a(n) changes sign infinitely often.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
Partial sums of A334659.
G.f. A(x) satisfies x = Sum_{k>=1} k^3 * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023
Sum_{k=1..n} k^3 * a(floor(n/k)) = 1. - Seiichi Manyama, Apr 03 2023
MATHEMATICA
Array[Sum[MoebiusMu[k]*k^3, {k, #}] &, 41] (* Michael De Vlieger, Jul 15 2020 *)
Accumulate[Table[MoebiusMu[n] n^3, {n, 50}]] (* Harvey P. Dale, Aug 15 2024 *)
PROG
(PARI) a(n) = sum(k=1, n, moebius(k)*k^3); \\ Michel Marcus, Jul 15 2020
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A336277(n):
if n <= 1:
return 1
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A336277(k1)
j, k1 = j2, n//j2
return c-((n*(n+1))**2-((j-1)*j)**2>>2) # Chai Wah Wu, Apr 04 2023
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Donald S. McDonald, Jul 15 2020
STATUS
approved