On the periodicity of some Farhi arithmetical functions

Details On the periodicity of some Farhi arithmetical functions On the periodicity of some Farhi arithmetical functions

2009-03-06

arxiv.org/abs/0903.1162v3

Mathematics

Number Theory

14 page

Scientific paper

Let $k\in\mathbb{N}$. Let $f(x)\in \Bbb{Z}[x]$ be any polynomial such that $f(x)$ and $f(x+1)f(x+2)... f(x+k)$ are coprime in $\mathbb{Q}[x]$. We call $$g_{k,f}(n):=\frac{|f(n)f(n+1)... f(n+k)|}{\text{lcm}(f(n),f(n+1),...,f(n+k))}$$ a Farhi arithmetic function. In this paper, we prove that $g_{k,f}$ is periodic. This generalizes the previous results of Farhi and Kane, and Hong and Yang.

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