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Further Symmetry In Primes

Simple geometry: from even numbers to primes

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

The first set A uses primes:

 p' = {x | 3p - 2}; p is prime

Is the output of (3p - 2) always a prime?

The second set B uses positive integers:

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

Using primes k = p that make true both conditions for set A and set B results in this interesting sequence of integers (all primes):

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

Now, the ratio p"/p' produces the plot below, suggesting that as p=k approaches infinity, there will be a p" prime that is 2p (???)

The largest prime I have tested is:

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

with ratio: 2.00000086256246

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