A366889 Dirichlet inverse of the highest power of two that divides the sum of divisors of n.

1, -1, -4, 0, -2, 4, -8, 0, 15, 2, -4, 0, -2, 8, 8, 0, -2, -15, -4, 0, 32, 4, -8, 0, 3, 2, -64, 0, -2, -8, -32, 0, 16, 2, 16, 0, -2, 4, 8, 0, -2, -32, -4, 0, -30, 8, -16, 0, 63, -3, 8, 0, -2, 64, 8, 0, 16, 2, -4, 0, -2, 32, -120, 0, 4, -16, -4, 0, 32, -16, -8, 0, -2, 2, -12, 0, 32, -8, -16, 0, 272, 2, -4, 0, 4, 4, 8, 0

Offset: 1

Author

Antti Karttunen, Jan 03 2024

Keywords

• sign
• mult

Comments

Multiplicative because A082903 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A082903, A336937, A359548, A359549 (parity of terms).

Programs

PARI

A082903(n) = (2^valuation(sigma(n), 2));
memoA366889 = Map();
A366889(n) = if(1==n,1,my(v); if(mapisdefined(memoA366889,n,&v), v, v = -sumdiv(n,d,if(dA082903(n/d)*A366889(d),0)); mapput(memoA366889,n,v); (v)));

Python

from functools import lru_cache
from sympy import divisor_sigma, divisors
@lru_cache(maxsize=None)
def A366889(n): return 1 if n==1 else -sum((1<<(~(m:=int(divisor_sigma(d))) & m-1).bit_length())*A366889(n//d) for d in divisors(n,generator=True) if d>1) # Chai Wah Wu, Jan 03 2024

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA082903(n/d) * a(d).