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Define a non-negative integer sequence $\{\mathcal{F}_n\}$ as follows: start with 1 and, at each step, insert the number of entries already present in the sequence which are factors of the last one.

This yields: $$1,1,2,3,3,4,4,5,3,5,4,6,7,3,6,9,7,4,7,5,5,6,10,8,8,9,8,10,9,9,10,10,11,3,\dots $$ My question basically is:

What the heck is this?

More formally, it is quite easy to show that $\{\mathcal{F}_n\}$ is unbounded. But other than that, I can see little that can be said trivially on the sequence. The first questions that come to my mind are:

  1. Is it true that every natural number appears in $\mathcal{F}$?
  1. How fast does $\mathcal{M}_n:=\max_{j<n}\{{\mathcal{F}_j}\}$ diverge?
  1. What's the subset of natural numbers whose elements have positive asymptotic density in $\mathcal{F}$?

The sequence is indexed on Sloane's Encyclopedia at: https://oeis.org/A124056.

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    $\begingroup$ Clearly, $\mathcal M_n\geq \sqrt{n-1}$: if none of the first $n-1$ terms is greater than $\sqrt{n-1}$, then at least one number appears there $\geq \sqrt{n-1}$ times; the last one would be followed by a large term. $\endgroup$
    – Ilya Bogdanov
    Commented Nov 20, 2020 at 9:07
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    $\begingroup$ Maybe it's also worth pointing out the even more trivial bound $\mathcal{M}_n< n$. Also notice: $$\text{every prime appears in the sequence}\implies$$ $$3\ \text{appears infinitely many times}\implies$$ $$\text{every natural number appears in the sequence}$$ $\endgroup$
    – Alessandro Della Corte
    Commented Nov 20, 2020 at 12:27

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