In a program I'm writing I'm using that the function:


$rphi(1) = 0$


$rphi(n) = 1+rphi(phi(n))$


grows very slowly. Judging from <https://oeis.org/A003434> it would seam like it is approximately logarithmic.


I was wondering if there are any known bounds on this function? Oeis didn't mention any.



Repetitions of the totient
This is a result of Pillai. Indeed we have $\text{rphi}(n)=\frac{\log n}{\log 2}$ when $n$ is a power of 2, and we have $\text{rphi}(n)=\lceil\frac{\log n}{\log 3}\rceil$ when $n$ is twice a power of 3. Pillai proved that these two cases characterise the extremal behaviour of $\text{rphi(n})$. 


$$\lceil\frac{\log n}{\log 2}\rceil\geq\text{rphi}(n)\geq\lceil\frac{\log n}{\log 3}\rceil$$
If one considers the numbers $n=2^a3^b$ one can see that $\frac{\text{rphi}(n)}{\log n}$ is dense in $[\frac{1}{\log 3},\frac{1}{\log 2}]$.



> 
> "On a function connected with ϕ(n)" S. S. Pillai, Bull. Amer. Math. Soc. Volume 35, Number 6 (1929), 837-841. 
> 
> 
> 


