Primes

We all know about prime numbers. I was just thinking about them, and one question came up.

What about primes that occur at prime(th) location in the list of prime numbers? And, how about that location also occurring at prime(th) location and so on..

For example, 7 is a prime number; it is 4th prime number after 2, 3 and 5. But 4 is not a prime so, 7 would not figure in that list. But 5 will work, because 5 is 3rd prime.

i.e., find all prime numbers that are 'positionally recursive primes'. Let p be a prime and pos(p) denote the position of p in the list of primes.

Then, we want all p such that
pos(p), pos(pos(p)),...
and so on.. are all primes.

It seems that this list can be generated. The smallest prime is 2 but it is 1st prime. So, 2 is not the candidate. 3 is 2nd prime and 2 itself is prime. Next number in the sequence would be 5 (3rd prime). Next will be 11 (5th prime) and so on...

Here are few initial numbers in the list:
3
5
11
31
127
709
5381
52711
648391
9737333
174440041
3657500101
88362852307
2428095424619
...


Now, an amazing co-incidence is that 5381 is my extension number.