On the use of the least common multiple to build a prime-generating recurrence

We study a recursively defined sequence which is constructed using the least common multiple. It has been conjectured that every term of that sequence is $1$ or a prime. In this paper we show that this claim is connected to a strong version of Linnik's Theorem, which is yet unproved. We also study a generalization on which composite numbers may appear depending on the initial term.