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

TL;DR

This paper investigates the distribution of numbers of the form k plus a function of k, specifically for divisor, prime divisor, and Euler's totient functions, establishing bounds on their counts up to x.

Contribution

It provides new bounds and asymptotic estimates for the number of integers up to x that can be expressed as k plus a specific arithmetic function of k.

Findings

Number of integers of the form k + ω(k) is at least proportional to x.

Number of integers of the form k + τ(k) is at most 0.94x.

Number of integers of the form k + φ(k) is at most 0.93x.

Abstract

For a function $f\colon \mathbb{N}\to\mathbb{N}$, 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, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and $\varphi(n)=\#\{1\leq k\leq n: \gcd(k,n)=1 \}$ be Euler's totient function. We show that \begin{align*} &(1) \quad x \ll N^+_{\omega}(x), \\ &(2) \quad x\ll N^+_{\tau}(x) \leq 0.94x, \\ &(3) \quad x \ll N^+_{\varphi}(x) \leq 0.93x. \end{align*}