On the periodicity of some Farhi arithmetical functions

Abstract

Let $k\in \mathbb{N}$. Let $f(x)\in \mathbb{Z}[x]$ 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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