Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that
$$\mathcal{B}=\sum_{n\geq 1}\left(\frac{1}{p_n}+\frac{1}{2+p_n}\right)<\infty,$$ (it is close to $1.9021...$) following a comment in MathWolrd, the page dedicated to the Brun's constant it expresses the scarcity of twin primes.
I think about the following idea, perhaps there is one better than the following attempt at exploration of this fact. If this question doesn't capture some of this essence (the exploration of the problem involving twin primes) you are welcome to add a comment.
We assume that there are infinitely many twin primes, as hypothesis to work (thus we assume the known asymptotic related in the so called Twin Prime Conjecture). For each integer $N\geq1$ we define the partial sum of previous series as $$\mathcal{B}_N=\sum_{n= 1}^{N}\left(\frac{1}{p_n}+\frac{1}{2+p_n}\right)=\frac{a_N}{b_N}$$
without a simplification of common factors (this is, in the case that $a_N$ and $b_N$ have common prime factors, we don't simplify these), thus we take $b_N=\prod_{k=1}^Np_k(2+p_k)$, and $a_N$ the corresponding finite series such that $\mathcal{B}_N=a_N/b_N$.
Example 1. As $\mathcal{B}_1=\frac{1}{3}+\frac{1}{5}=\frac{5+3}{3\cdot5}$, then we take $a_1=8$ and $b_2=15$.
Example 2. (Update.) See this technical details in comments, for the first interpretation, $$\mathcal B_2=\frac13+\frac15+\frac15+\frac17=\frac{5\cdot7+2\cdot3\cdot7+3\cdot5}{3\cdot5\cdot7}.$$
Important. Thus we take this interpretation (the first interpretation in comments).
Now we take $N$ large enough to ensure that $a_N>b_N$ (after the first terms in the sequence $(\mathcal{B}_N)_{N\geq 1}$, we have this assert).
My idea to explore is use the prime counting function $\pi(x)$ and ask
Question. When $N\to\infty$, what can you say about an asymptotic for $$\pi(a_N)-\pi(b_N)?$$ I say an asymptotic with an error term (small or big oh) or well an equivalence $\sim$.
Thanks in advance.