Let $a\left(m,n\right)$ be $n$-th full reptend prime in base $m$, where $\left|m\right|\geq2$.
Does the series $\displaystyle \sum_{n=1}^\infty a\left(m,n\right)$ converge?
For example,
$\displaystyle \sum_{n=1}^\infty a\left(10,n\right) = \dfrac{1}{7}+\dfrac{1}{17}+\dfrac{1}{19}+\dfrac{1}{23}+\dfrac{1}{29}+\dfrac{1}{47}+...$
P.S : List of full reptend primes in various base is provided at https://en.wikipedia.org/wiki/Full_reptend_prime#Full_reptend_primes_in_various_bases