This is the "Cunningham chain of the second kind of length 2", here is the <a href="http://oeis.org/A005382">OEIS reference</a> and here is the <A HREF="https://en.wikipedia.org/wiki/Cunningham_chain">Wikipedia entry</A>; record primes of this type are listed <A HREF="https://primes.utm.edu/top20/page.php?id=20">here</A>, the currently largest known prime $p$ such that $2p-1$ is also prime is $p=129431439657 \cdot 2^{170172} + 1$. It is believed that the number of primes of this type less than $N$ grows with bound as $1.32\, N/\log^2 N$.