  For a function $f\colon \mathbb{N}\to\mathbb{N}$, define
$N^{\times}_{f}(x)=\#\{n\leq x: n=kf(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 $\varphi(n)=\#\{1\leq k\leq n: (k,n)=1 \}$ 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
$c_0=1.365...$\,.
