We study a recursively defined sequence which is constructed using the least common multiple. Several authors have 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 still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers.