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a(n) = Sum_{k=1..n} mu(k)*k^3.
8

%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