$127$ has an interesting property: It is the $31$st prime number and its rank ($31$) is also a prime. $31$ is the $11$th prime so its rank is also a prime. $11$ is also a prime number with a rank ($5$) that is also a prime. $5$ is the 3rd prime number and and so its rank ($3$) is also a prime. And finally $3$ is the $2$nd prime so its rank is also a prime... Is there a name for primes whose rank (index in the prime series) is also a prime? That is:
If $a_i$ is the $i$th prime number, then $i$ is a also a prime.
How about numbers like $127$ where going down ranks of ranks always produce prime numbers (down to rank $2$ of course)? The first $11$ :-) primes in this (infinite) series would be:
$3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, \dotsc$