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

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 ${\sum }_{n\le x}\log a_n$.

7 pages; This research was initiated by Laurenţiu Panaitopol (1940--2008), former professor at the Faculty of Mathematics, University of Bucharest, Romania. The present paper is dedicated to his memory