This thread got me thinking...

Define $\displaystyle \varphi_0(n)=n,\; \varphi_1(n)=\varphi(n),\; \varphi_2(n)=\varphi(\varphi(n)) $ and so on...

Now Define $\displaystyle I(n) $ to be the smallest integer such that $\displaystyle \varphi_{I(n)}(n)=1 $.

I've found some bounds of $\displaystyle I(n) $: it's easy to see $\displaystyle I(n)\leq n-1 $ and a delicate strong induction shows $\displaystyle I(n)\leq\varphi(n) $.

My question is, what the formula for $\displaystyle I(n) $, or at least what are some tighter bounds?

-Thanks!