On Rowland's Sequence

Abstract

E. S. Rowland proved that $a_k= a_{k-1}+ \gcd(k,a_{k-1})$, $a_1= 7$ implies that $a_{k}-a_{k-1}$ is always 1 or prime. Conjecturally this property also holds for any $a_1>3$ from a certain $k$ onwards. We state some properties of this sequence for arbitrary values of $a_1$. Namely, we prove that some specific sequences contain infinitely many primes and we characterize the possible finite subsequences of primes.