cp's OEIS Frontend

A336277 a(n) = Sum_{k=1..n} mu(k)*k^3.

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