**PART I** (Initial version)


Let   $P$   be the set of all primes   $2\ 3\ \ldots$.   Let


$$P\_d\ \ :=\ \ \{\ p\in P\ :\ \exists\_{q\in P}\ \ 0 < |p-q|\le d\ \}$$


and


$$S\_d\ :=\ \sum\_{p\in P\_d}\ \frac 1p$$


for every real   $d>0$.   Thus   $d\mapsto S\_d$   is non-decreasing,   $S\_2 < \infty$,   and   $\lim\_{d\rightarrow\infty} S\_d = \infty$.   What else is known about   $S\_d$ ?   For which values of   $d$   the sum   $S\_d$   is finite?


**PART II** (additional)


Let   $d\ m\ n$   be positive integers. Is it true that


$$ (m < n)\quad\Rightarrow\quad \left( S\_{m+d}-S\_m\ \ge\ S\_{n+d}-S\_n \right)$$


**?** - It seems (to me) that it should be true.



Are sums of the inverses of prime siblings finite?
It turns out that $S\_d$ is finite for every $d$. See my answer to this [very similar question](https://math.stackexchange.com/questions/397391/convergence-of-the-sum-of-reciprocals-of-fake-twin-primes/397466#397466).


