Consider the function $F:\mathbb{N}\to\mathbb{N}$ such that $F(n)=\tfrac{n^2-n}{\delta(n^2-n)}$, where $\delta$ returns the biggest prime factor of its input. I wonder if this function always stabilizes for any $n\ge2$, e.g.
$$41\to40\to120\to840\to840$$
Notice that the sequence $\{F^{(k)}(n)\}_k$ will always stabilize whenever $F^{(k)}(n)-1$ is prime.
After running some code, it looks like this happens quite often but perhaps not always, e.g.
$$116\to460\to9180\to615060\to69501780\to3098458854180$$ $$\to228250571100148357020\to1597753997701038499140$$ $$\to66376869992686262963920779540\to3051942105393721684818113522469660$$ $$\to1842552059231456986496924495034130301460$$ $$\to14625963786809032272119228874352339583592033747278816254956293171340\to?$$
after which the program crashes. I wonder if there is a rigorous way to justify whether this function always stabilizes or not.
