On the iterates of shifted Euler's function

Abstract

Let $\varphi$ be the Euler's function and fix an integer $k\ge 0$. We show that, for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi(x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for every initial value $x_1,x_2\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi(x_{n+1})+\varphi(x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that $k$ is even.