Let the map $T: \mathbb N \rightarrow \mathbb N$ be defined as follows:
Take the prime factors of the input number (each factor only once, regardless of multiplicity), order them by magnitude (ascending) and concatenate their decimal representations together to form the output number.
Example: $36$ has prime factorization $2^2 \times 3^2$, so $T(36) = 23$.
$T(n)$ can be less than, equal to, or greater than $n$. For instance
- $T(100) = 25$
- $T(99) = 311$
- $T(7) = 7$
Collatz-like questions can now be asked about iterating $T$ on the natural numbers. In particular
Are there any non-trivial cycles, i.e. numbers $n$ such that $T(n) \ne n$ but there is an iteration $T^{(k)}(n) = n$ with $k > 1$?
Are there numbers $n$ for which iterating $T$ grows without bounds?
Experimentally, the vast majority of numbers turn into a prime number (and thus a trivial cycle) after just two or three applications of $T$. But there are some numbers ($91$ being the smallest example) for which the orbit of $T$ is still growing after $50$ iterations (by which time it reaches $10^{65}$ and factorization becomes expensive) so the answer to 2 might be "Yes".