MathNet 2019-part-04 — 184 problems (EN–VI) Question 101

Do there exist a positive integer $k$ and a non-constant sequence $a_{1}, a_{2}, a_{3}, \ldots$ of positive integers such that $a_{n}=\operatorname{gcd}\left(a_{n+k}, a_{n+k+1}\right)$ for all positive integers $n$?