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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:

  1. This sequences forms a subsequence of the primes, are there any useful bounds on the density of this sub-sequence?

  2. How does the $n$th nice prime compare in size to the $n$th prime?

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  • $\begingroup$ We have $\lim\limits_{n\to \infty}\frac{p_n}{n\log(n)}=1$ $\endgroup$ – Jorge Fernández Hidalgo Jul 29 '16 at 13:58
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The sequence of nice prime is the following : $$ n_i = p_{p_i} \ \ \ \text{for}\ i = 1,2,... $$ Where $p_k$ is the $k$-th standard prime number.

Using the approximation $p_k \approx k \ln(k)$, we can estimate $n_i \approx p_{i \cdot \ln(i)} \approx i\ \ln(i) \cdot \ln(i \ \ln(i))$

See WA for a small table (starting at $i=2$).

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