On the periodicity of some Farhi arithmetical functions

Qing-Zhong Ji, Chun-Gang Ji

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+…