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 bounded but that the union of all such sets is unbounded.

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.