On the periodicity of some Farhi arithmetical functions

Let $k\in\mathbb{N}$. Let $f(x)\in \mathbb{Z}[x]$ be any polynomial such that $f(x)$ and $f(x+1)f(x+2)\cdots f(x+k)$ are coprime in $\mathbb{Q}[x]$. We call

$$g_{k,f}(n):=\frac{\vert f(n)f(n+1)\cdots f(n+k)\vert} {\text{lcm}(f(n),f(n+1),\cdots,f(n+k))}$$

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.