%I #33 May 04 2023 01:55:56

%S 1,-3,-12,-12,-37,-1,-50,-50,-50,50,-71,-71,-240,-44,181,181,-108,

%T -108,-469,-469,-28,456,-73,-73,-73,603,603,603,-238,-1138,-2099,

%U -2099,-1010,146,1371,1371,2,1446,2967,2967,1286,-478,-2327,-2327,-2327,-211,-2420

%N a(n) = Sum_{k=1..n} mu(k)*k^2.

%C Conjecture: a(n) changes sign infinitely often.

%H Seiichi Manyama, <a href="/A336276/b336276.txt">Table of n, a(n) for n = 1..10000</a>

%F Partial sums of A334657.

%F G.f. A(x) satisfies x = Sum_{k>=1} k^2 * (1 - x^k) * A(x^k). - _Seiichi Manyama_, Apr 01 2023

%F Sum_{k=1..n} k^2 * a(floor(n/k)) = 1. - _Seiichi Manyama_, Apr 03 2023

%t Array[Sum[MoebiusMu[k]*k^2, {k, #}] &, 47] (* _Michael De Vlieger_, Jul 15 2020 *)

%o (PARI) a(n) = sum(k=1, n, moebius(k)*k^2); \\ _Michel Marcus_, Jul 15 2020

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A336276(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)*((j2<<1)-1)-j*(j-1)*((j<<1)-1))//6*A336276(k1)

%o j, k1 = j2, n//j2

%o return c-(n*(n+1)*((n<<1)+1)-j*(j-1)*((j<<1)-1))//6 # _Chai Wah Wu_, Apr 04 2023

%Y Cf. A002321, A068340, A336277, A336278, A336279.

%Y Cf. A008683, A055615, A070891, A360390, A361983.

%K easy,sign

%O 1,2

%A _Donald S. McDonald_, Jul 15 2020