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

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 $(a_n){n\ge 1}$,andproveasymptoticresultsforthesums$\sum{n\le x} a_n$,$\sum_{n\le x} 1/a_n$and$\sum_{n\le x} \log a_n$.