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A353366
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Dirichlet inverse of A110963, which is a fractalization of Kimberling's paraphrases sequence (A003602).
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4
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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
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OFFSET
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1,5
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LINKS
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FORMULA
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a(1) = 1; a(n) = -Sum_{d|n, d < n} A110963(n/d) * a(d).
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PROG
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(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003602(n) = (1+(n>>valuation(n, 2)))/2;
v353366 = DirInverseCorrect(vector(up_to, n, A110963(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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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