[math-fun] Iteration question

Let f(n) = sigma(n) - n = sum of proper divisors of n. If you iterate f on positive integer n, some trajectories end at 0 (where you cannot continue, since 0 is not in the domain of f), some trajectories loop (e.g. 6 -> 6... or 220 -> 284 -> 220...). For other numbers, like 138, the trajectory seems to diverge. Can we prove that there exist divergent trajectories?