Numbers of the form $kf(k)$

For a function $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$, define N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for somek} \}. 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 $\phi (n)=\mathrm{\#}\{1\le k\le n:(k,n)=1\}$ be Euler’s totient function. We prove that \begin{gather*} !!!!!!!!!!!!!!!!!!!!!!! 1) \quad N^{\times}{\tau}(x) \asymp \frac{x}{(\log x)^{1/2}}; \ 2) \quad N^{\times}{\omega}(x) = (1+o(1))\frac{x}{\log\log x}; \ !!!!!!!!! 3) \quad N^{\times}_{\varphi}(x) = (c_0+o(1))x^{1/2}, \end{gather*} where $c_0=\mathrm{1.365...}$,.