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

For a function $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$, let $$N_f^{+}(x)=\{n\le x:n=k+f(k)forsomek\}\mathrm{.}$$ Let $\tau (n)={\sum }_{d\mathrm{\mid }n}1$ be the divisor function, $\omega (n)={\sum }_{p\mathrm{\mid }n}1$ be the prime divisor function, and $\phi (n)=\mathrm{\#}\{1\le k\le n:\gcd (k,n)=1\}$ be Euler’s totient function. We show that \begin{align*} &(1) \quad x \ll N^+_ω(x), \ &(2) \quad x\ll N^+τ(x) \leq 0.94x, \ &(3) \quad x \ll N^+{\varphi}(x) \leq 0.93x. \end{align*}