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Further Symmetry In Primes

Two disjoint (?) sets of primes can be generated using two distinct rules:

The first set P' contains primes:

 P' = {x | x = P(3p - 2)}; p is prime and P(.) is prime

(Is x = (3p - 2) always a prime?)

The second set P" uses natural numbers n :

 P" = {x | x = P(6n - 1)}; P(.) is prime

Using primes p = n results in a sequence of primes:

3,5,7,23,43,47,53,67,103, ... , ?

The ratio P"/P' produces the plot below.

The largest prime tested is:

p = 1159337; p' = 3478009; p" = 6956021

with ratio: 2.00000086256246