On numbers not representable as $n + w(n)$

TL;DR

This paper investigates the distribution of numbers of the form n + w(n), where w(n) is an additive function, focusing on their modular properties and the density of non-representable numbers.

Contribution

It provides new bounds on the density of numbers that cannot be expressed as n + w(n), advancing understanding of additive functions and their additive properties.

Findings

Lower bound for the density of non-representable numbers.

Distribution analysis of n + w(n) modulo p.

Insights into additive functions on primes.

Abstract

Let $w(n)$ be an additive non-negative integer-valued arithmetic function which is equal to $1$ on primes. We study the distribution of $n + w(n)$ $\pmod p$ and give a lower bound for the density of the set of numbers which are not representable as $n + w(n)$.