Define $f(n) = \lfloor {ne}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/e}\rfloor$ if $n$ is even. Is the set $\{n, f(n), f(f(n)),\dots\}$ bounded for every $n$?
Computer sampling suggests that each such set is finite - indeed, that the iterates reach 0 - but that the set of cardinalities of the sets is infinite.
Write the iteration as $x(j+1) = f(x(j))$ with $x(0) = n$. Heuristically, while $x(j)$ is large, $\log x(j)$ undergoes a "random walk" with a slight bias to the left; with probability $1$ you eventually get to a small enough value of $x(j)$, and then you're trapped in a fixed point or cycle. This heuristic should not be taken too seriously, but it does suggest that the set should always be finite, although sometimes very large.
