%I #28 Aug 15 2024 13:14:32
%S 1,-7,-34,-34,-159,57,-286,-286,-286,714,-617,-617,-2814,-70,3305,
%T 3305,-1608,-1608,-8467,-8467,794,11442,-725,-725,-725,16851,16851,
%U 16851,-7538,-34538,-64329,-64329,-28392,10912,53787,53787,3134,58006,117325,117325,48404
%N a(n) = Sum_{k=1..n} mu(k)*k^3.
%C Conjecture: a(n) changes sign infinitely often.
%H Seiichi Manyama, <a href="/A336277/b336277.txt">Table of n, a(n) for n = 1..10000</a>
%F Partial sums of A334659.
%F G.f. A(x) satisfies x = Sum_{k>=1} k^3 * (1 - x^k) * A(x^k). - _Seiichi Manyama_, Apr 01 2023
%F Sum_{k=1..n} k^3 * a(floor(n/k)) = 1. - _Seiichi Manyama_, Apr 03 2023
%t Array[Sum[MoebiusMu[k]*k^3, {k, #}] &, 41] (* _Michael De Vlieger_, Jul 15 2020 *)
%t Accumulate[Table[MoebiusMu[n] n^3,{n,50}]] (* _Harvey P. Dale_, Aug 15 2024 *)
%o (PARI) a(n) = sum(k=1, n, moebius(k)*k^3); \\ _Michel Marcus_, Jul 15 2020
%o (Python)
%o from functools import lru_cache
%o @lru_cache(maxsize=None)
%o def A336277(n):
%o if n <= 1:
%o return 1
%o c, j = 1, 2
%o k1 = n//j
%o while k1 > 1:
%o j2 = n//k1 + 1
%o c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A336277(k1)
%o j, k1 = j2, n//j2
%o return c-((n*(n+1))**2-((j-1)*j)**2>>2) # _Chai Wah Wu_, Apr 04 2023
%Y Cf. A002321, A068340, A336276, A336278, A336279.
%Y Cf. A008683, A055615, A070891.
%K easy,sign
%O 1,2
%A _Donald S. McDonald_, Jul 15 2020