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Warning: This question contains non-standard language for the problem.

I define the collatz length $\operatorname{cl}(n)$ of an integer $n$ to be the minimum number of steps that takes for it to reach 1 by applying the procedure described in Collatz Conjecture. For example, $2 \rightarrow 1$ so, $\operatorname{cl}(2) = 1$, and $4 \rightarrow 2 \rightarrow 1$, so $\operatorname{cl}(4) = 2$.

I was trying to find integers for which $\operatorname{cl}(n) = n$. Using my computer, I checked all natural numbers up to $10^6$. Within that interval, the only one that satisfied the conditions was $5$. I am trying to understand whether there is any other number satisfying this condition, but since I am not familiar with the literature, it is hard for me to reach the existing tools. If you have a suggestion or comment about this question, then I would like to hear it.

Einsteinium
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Cahit Sertöz
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    In the following post they conjecture $cl(n) =\Theta(\log n)$, and even calculate the average of $cl$. If this conjecture is true then you would have only finitely many solutions to $cl(n)=n$. It might be that this might be showable without the Collatz conjecture but I wouldn't know https://math.stackexchange.com/questions/4553892/average-number-of-steps-in-the-collatz-conjecture – Bruno Andrades Apr 19 '25 at 02:39
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    This is the sequence https://oeis.org/A006577 – jjagmath Apr 19 '25 at 02:40
  • I want to mention that the conjecture I linked is strictly stronger than the Collatz conjecture. So I'm only saying that it seems reasonable (given the numerical data + Collatz) that this would be the case, I'm not giving any actual evidence of anything – Bruno Andrades Apr 19 '25 at 02:42

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