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

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

For a function $f:\mathbb{N}\to \mathbb{N}$, 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, $\omega (n)={\sum }_{p|n}1$ b…