Define a 'nice prime' by a prime such that it is the $n$th prime where $n$ is a prime number. For example, $3$ is a nice prime since it is the $2$nd prime and 2 is a prime itself. $5$ is also a nice prime since it is the $3$rd prime and $3$ is prime. $7$ is not a nice prime since it is the $4$th prime.
This leads to two natural (related) questions:
This sequences forms a subsequence of the primes, are there any useful bounds on the density of this sub-sequence?
How does the $n$th nice prime compare in size to the $n$th prime?