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:
- Is it true that every natural number appears in $\mathcal{F}$? Is it true that every natural appears infinitely many times?
- How fast does $\mathcal{M}_n:=\max_{j<n}\{{\mathcal{F}_j}\}$ diverge?
The sequence is indexed on Sloane's Encyclopedia at: https://oeis.org/A124056.