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A336278 a(n) = Sum_{k=1..n} mu(k)*k^4.

%I A336278 #31 Apr 04 2023 22:06:32
%S A336278 1,-15,-96,-96,-721,575,-1826,-1826,-1826,8174,-6467,-6467,-35028,
%T A336278 3388,54013,54013,-29508,-29508,-159829,-159829,34652,268908,-10933,
%U A336278 -10933,-10933,446043,446043,446043,-261238,-1071238,-1994759,-1994759,-808838,527498,2028123
%N A336278 a(n) = Sum_{k=1..n} mu(k)*k^4.
%C A336278 Conjecture: a(n) changes sign infinitely often.
%H A336278 Seiichi Manyama, <a href="/A336278/b336278.txt">Table of n, a(n) for n = 1..10000</a>
%F A336278 Partial sums of A334660.
%F A336278 From _Seiichi Manyama_, Apr 03 2023: (Start)
%F A336278 G.f. A(x) satisfies x = Sum_{k>=1} k^4 * (1 - x^k) * A(x^k).
%F A336278 Sum_{k=1..n} k^4 * a(floor(n/k)) = 1. (End)
%t A336278 Array[Sum[MoebiusMu[k]*k^4, {k, #}] &, 35] (* _Michael De Vlieger_, Jul 15 2020 *)
%t A336278 Accumulate[Table[MoebiusMu[x]x^4,{x,40}]] (* _Harvey P. Dale_, Jan 14 2021 *)
%o A336278 (PARI) a(n) = sum(k=1, n, moebius(k)*k^4); \\ _Michel Marcus_, Jul 15 2020
%o A336278 (Python)
%o A336278 from functools import lru_cache
%o A336278 @lru_cache(maxsize=None)
%o A336278 def A336278(n):
%o A336278     if n <= 1:
%o A336278         return 1
%o A336278     c, j = 1, 2
%o A336278     k1 = n//j
%o A336278     while k1 > 1:
%o A336278         j2 = n//k1 + 1
%o A336278         c -= (j2*(j2**2*(j2*(6*j2 - 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30*A336278(k1)
%o A336278         j, k1 = j2, n//j2
%o A336278     return c-(n*(n**2*(n*(6*n + 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30 # _Chai Wah Wu_, Apr 04 2023
%Y A336278 Cf. A002321, A068340, A336276, A336277, A336279.
%Y A336278 Cf. A008683, A055615, A070891, A344429, A344430.
%K A336278 easy,sign
%O A336278 1,2
%A A336278 _Donald S. McDonald_, Jul 15 2020