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*}