Numbers of the form $k+f(k)$

Abstract

For a function $f\colon \mathbb{N}\to\mathbb{N}$, let $$ N^+_f(x)=\{n\leq x: n=k+f(k) \mbox{ for some } k\}. $$ Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and $\varphi(n)=\#\{1\leq k\leq n: \gcd(k,n)=1 \}$ be Euler's totient function. We show that \begin{align*} &(1) \quad x \ll N^+_{\omega}(x), \\ &(2) \quad x\ll N^+_{\tau}(x) \leq 0.94x, \\ &(3) \quad x \ll N^+_{\varphi}(x) \leq 0.93x. \end{align*}