MathNet 2024-part-03 — 168 problems (EN–VI) Question 157

Find all sequences $a_1, a_2, \ldots$ of positive integers such that the expression $n a_n - m a_m + 2 a_m - 1$ is divisible by $a_n + a_m - 1$ for all $n, m \ge 1$.
(Unubold Munkhbat)