Iterated Pi function

Question

Does anyone have any information on iterating the prime counting function. Specifically, $\pi_n(x)$=$\pi(\pi_{n-1}(x))$, and $\pi_1(x)$=$\pi(x)$. I'm looking for anything on this function, what it may be called (when I search for iterate pi function, all I get is information about calculating pi).

Answer 1

$\pi_n(x)\sim\frac{x}{\log^nx}$ for any fixed $n$, though this estimate will be too large if $n$ is allowed to grow with $x$. You could get a better estimate by iterating the logarithmic integral function. Using this estimate I guess that $\pi_{10}(10^{100})\approx7.8\times10^{76}.$