Some properties of a sequence defined with the aid of prime numbers

TL;DR

This paper studies a sequence defined via prime numbers, analyzing its properties and deriving asymptotic formulas for sums involving the sequence and related functions.

Contribution

It introduces and investigates the sequence where each term is the smallest positive integer making the sum with its index prime, providing new asymptotic results.

Findings

Asymptotic formulas for sums of the sequence and related functions

Behavior of the sequence as n grows large

Connections to prime number distribution

Abstract

For every integer $n\ge 1$ let $a_n$ be the smallest positive integer such that $n+a_n$ is prime. We investigate the behavior of the sequence $(a_n)_{n\ge 1}$, and prove asymptotic results for the sums $\sum_{n\le x} a_n$, $\sum_{n\le x} 1/a_n$ and $\sum_{n\le x} \log a_n$.