A361982 a(n) = 1 + Sum_{k=2..n} (-1)^k * k * a(floor(n/k)).

1, 3, 0, 8, 3, -3, -10, 22, 22, 12, 1, -23, -36, -50, -35, 93, 76, 76, 57, 17, 38, 16, -7, -103, -103, -129, -129, -185, -214, -184, -215, 297, 330, 296, 331, 331, 294, 256, 295, 135, 94, 136, 93, 5, 5, -41, -88, -472, -472, -472, -421, -525, -578, -578, -523, -747, -690, -748

Offset: 1

Author

Seiichi Manyama, Apr 02 2023

Keywords

• sign
• look

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..8191

Crossrefs

Partial sums of A332793.

Cf. A347030, A361983.

Cf. A068340.

Programs

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A361982(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (((j2<<1)-1 if j2&1 else -(j2<<1)+1)+(-(j<<1)+1 if j&1 else (j<<1)-1)>>2)*A361982(k1)
        j, k1 = j2, n//j2
    return c+((-(n<<1)-1 if n&1 else (n<<1)+1)+(-(j<<1)+1 if j&1 else (j<<1)-1)>>2) # Chai Wah Wu, Apr 02 2023

Formula

Sum_{k=1..n} (-1)^k * k * a(floor(n/k)) = -1.

G.f. A(x) satisfies -x = Sum_{k>=1} (-1)^k * k * (1 - x^k) * A(x^k).