Numbers of the form $kf(k)$

Mikhail R. Gabdullin, Vitalii V. Iudelevich, Florian Luca

For a function $f:\mathbb{N}\to \mathbb{N}$, 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, $ω(n)=\su…