Let us consider the following operation on positive integers:  $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ Is it true that if we apply this operation to any integer multilpe times, it will eventually get into a finite cycle? Is there a constant $K$ such that any integer will fall into a cycle after $K$ steps?

I think that the answer to the first question is yes, but to the second no. We tested the first $10000$ integers and every integer fell into a cycle after at most $6$ steps.

**Edit:** @MarkSapir proved that the answer to the second question is no. His proof raises the (third) question: How long can such a cycle be?