Introduction to Computers : IDC101

Let me pose a problem here. I am not sure if it is there on Project Euler, but I am sure Project Euler has many problems which have similar taste.

Consider the function $\pi$, where $\pi(n)$ is the number of primes $\leq n$. Denote $\pi^2(n) = \pi(\pi(n))$, $\pi^3(n) = \pi(\pi(\pi(n)))$, and so on. This gives a strictly decreasing sequence, and hence $\pi^r(n) = 2$ for some $r$. This $r$ depends on $n$. I would take a liberty to call this $r$ the $\pi$-length of $n$ and the sequence $\pi(n), \pi^2(n), \pi^3(n), \cdots, \pi^2(n) = 2$, the $\pi$-sequence of $n$?

Now I can ask several questions.

1. Find a prime number whose $\pi$-sequence consists only of prime numbers? Larger the better!
2. Find a natural number whose $\pi$-sequence contains no prime other than 2. Again, larger the better.
   
3. Among the natural numbers $\leq 10000$, which one has largest $\pi$-length?
4. Plot the function $\pi$-length.

I am sure you may also ask plenty of questions based on this.

Enjoy Pythoning!