2-Digit Numbers forming Longest Reverse-and-Add Sequence

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Theorem

Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.

Let $r \left({m}\right)$ be the reverse-and-add process on $m$.

Let $r$ be applied iteratively to $m$.


The $2$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are $89$ and $98$, both needing $24$ iterations.


Proof

The sequence obtained by iterating $r$ on $89$ is:

$89, 187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
$47267087164, 93445163438, 176881317877, 955594506548, 170120002107, 8713200023178$

This sequence is A033670 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Note that $r \left({89}\right) = r \left({98}\right) = 187$, so the sequence obtained by iterating $r$ on $98$ is the same.



Sources