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).

Searching the OEIS gets you this although there doesn't appear to be much information. — Jack M, Aug 13 '14 at 00:11
@JackM: Also A007097, where a(n) is the least number such that $\pi_n(a(n))=1.$ — Charles, Aug 13 '14 at 03:16

score 2 · Answer 1 · answered Aug 13 '14 at 03:12

2

$\pi_n(x)\sim\frac{x}{\log^nx}$ 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 $\pi_{10}(10^{100})\approx7.8\times10^{76}.$

answered Aug 13 '14 at 03:12

Charles

32,122

1

Using the Prime Number Theorem also gives an extremely bad estimate when $\pi^n(x)$ is small. – vukov Aug 13 '14 at 15:57
1

@vukov: Agreed. That's another manifestation of $n$ growing with $x$. – Charles Aug 13 '14 at 18:20

Iterated Pi function

1 Answers1

Linked