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A336276
a(n) = Sum_{k=1..n} mu(k)*k^2.
9
1, -3, -12, -12, -37, -1, -50, -50, -50, 50, -71, -71, -240, -44, 181, 181, -108, -108, -469, -469, -28, 456, -73, -73, -73, 603, 603, 603, -238, -1138, -2099, -2099, -1010, 146, 1371, 1371, 2, 1446, 2967, 2967, 1286, -478, -2327, -2327, -2327, -211, -2420
OFFSET
1,2
COMMENTS
Conjecture: a(n) changes sign infinitely often.
LINKS
FORMULA
Partial sums of A334657.
G.f. A(x) satisfies x = Sum_{k>=1} k^2 * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023
Sum_{k=1..n} k^2 * a(floor(n/k)) = 1. - Seiichi Manyama, Apr 03 2023
MATHEMATICA
Array[Sum[MoebiusMu[k]*k^2, {k, #}] &, 47] (* Michael De Vlieger, Jul 15 2020 *)
PROG
(PARI) a(n) = sum(k=1, n, moebius(k)*k^2); \\ Michel Marcus, Jul 15 2020
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A336276(n):
if n <= 1:
return 1
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c -= (j2*(j2-1)*((j2<<1)-1)-j*(j-1)*((j<<1)-1))//6*A336276(k1)
j, k1 = j2, n//j2
return c-(n*(n+1)*((n<<1)+1)-j*(j-1)*((j<<1)-1))//6 # Chai Wah Wu, Apr 04 2023
KEYWORD
easy,sign
AUTHOR
Donald S. McDonald, Jul 15 2020
STATUS
approved