Details:
The following terminology is non-standard.
Definition 1: A reverse pair of numbers is a pair of numbers $m$ and $n$ such that, if the decimal expansion of $m$ is $$\overline{a_1a_2\cdots a_k},$$ then that of $n$ is $$\overline{a_ka_{k-1}\cdots a_2a_1},$$ where $k\ge 2$.
For example, $35$ and $53$ form such a pair.
Definition 2: A string of the iterated Collatz conjecture function is a word $w$ in the alphabet $\Bbb N$ such that its alphabet members are exclusively in the order from left to right according to the Collatz conjecture function.
For example, $(5, 16, 8, 4, 2, 1)$ is such a string.
The Question:
I'm looking for numbers $N$ whose strings (starting with $N$) of the iterated Collatz conjecture function contain reverse pairs. Are they significant? How "common" are they?
My attempt:
I found that $(35, 106, 53, \text{etc.})$ is such a string, so $35$, its double $70$, and so on, are examples. The pair $35$ and $53$ inspired this question.
I merely guess a search using some programme or another for such $N$s would yield diminishing results as candidate $N$s increase. This guess is based on my own difficulty conjuring them up in my head or on paper.
Palindromic numbers stop the iteration of the function for obvious reasons. I'll leave it to you whether or not that's cheating, though, and whether we need to make sure $m\neq n$.
Please help :)
Disclaimer: I'm not trying to examine the Collatz conjecture. I'm just curious. I am aware that this is very "base ten" in nature.
