Just a little empirical data up to $N=10^{10}$,
three superimposed random runs.
Growing roughly linearly w.r.t. $\log_{10} N$ within that range.
<hr/>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://i.sstatic.net/BYqot.jpg" width="500" />


  [1]: https://i.sstatic.net/BYqot.jpg