A353366 Dirichlet inverse of A110963, which is a fractalization of Kimberling's paraphrases sequence (A003602).

1, -1, -1, 0, -2, 1, -1, 0, -2, 2, -2, 0, -4, 1, 3, 0, -5, 2, -3, 0, -4, 2, -2, 0, -3, 4, 1, 0, -8, -3, -1, 0, -5, 5, -1, 0, -10, 3, 5, 0, -11, 4, -6, 0, -4, 2, -2, 0, -12, 3, 3, 0, -14, -1, 4, 0, -9, 8, -8, 0, -16, 1, 14, 0, -1, 5, -9, 0, -14, 1, -5, 0, -19, 10, -4, 0, -16, -5, -3, 0, -12, 11, -11, 0, -2, 6, 10

Offset: 1

Author

Antti Karttunen, Apr 18 2022

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A003602, A110963, A353367.

Cf. also A349134, A353368.

Programs

PARI

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
A353366(n) = v353366[n];

Python

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A353366(n): return 1 if n==1 else -sum(((1+(m:=d>>(~d&d-1).bit_length())>>(m+1&-m-1).bit_length())+1)*A353366(n//d) for d in divisors(n,generator=True) if d>1) # Chai Wah Wu, Jan 04 2024

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A110963(n/d) * a(d).

a(n) = A353367(n) - A110963(n).