I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' theorems or Brun's theorem. On the other hand I know from an informative point of view that are in the literature other statements and/or conjectures more complicated about the convergence of series involving prime numbers, see for example the first paragraph of section E7 from .
Motivation (Edited, thanks for the comments). I am going to ask just for the following prime constellation with the goal that my question isn't too broad, but I think that to focus the attention about the sequence of strong primes is interesting itself, the definition is the definition in number theory of Wikipedia Strong prime. Also I would like to know your good intuition/reasoning to get statements or conjectures of this kind.
Question. What can be interesting series involving the sequences of strong primes? I am asking about statements or conjectures concerning convergent or divergent series involving strong primes. I am asking about series with a good mathematical content or significance in mathematics. Many thanks.
I don't know if some series involving such strong primes are in the literature. If this question is welcome, feel free to add the reference and I try to find and read those results from the literature, in other case add your genunine series for strong primes.
 Richard K. Guy, Unsolved Problems in Number Theory, Unsolved problems in Intuitive Mathematics Volume I, Second Edition, Springer-Verlag (1994).
 I add also that the encyclopedia MarthWolrd has the article Prime Sums.