See
https://en.wikipedia.org/wiki/Conway_chained_arrow_notation
for the details how conway chained arrow notation works.
- I want to calculate the approximate value $n$ such that
$$n\rightarrow n\rightarrow n\rightarrow n<a\rightarrow a\rightarrow a\rightarrow a\rightarrow a<(n+1)\rightarrow (n+1)\rightarrow (n+1) \rightarrow(n+1)$$
for $a\ge 3$
The number $n\rightarrow n\rightarrow ... \rightarrow n$ with $n$ $n's$ is approximately $\large f_{\omega^2}(n)$ (See https://en.wikipedia.org/wiki/Fast-growing_hierarchy for details of the fast growing hierarchy), but this does not seem to help very much.
Is it true that $n$ must exceed $a\rightarrow a\rightarrow a\rightarrow a$ ? I think it is, but I do not know how I can prove it.