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 (52138 digits) prime $p$ such that $2p-1$ is also prime is $p=129431439657 \cdot 2^{170171} + 1$. It is a (still open) conjecture that the number of primes of this type less than $N$ grows with bound as $1.32\, N/\log^2 N$, see <A HREF="http://ac.inf.elte.hu/Vol_038_2012/117_38.pdf">*Sieving for large twin primes and Cunningham chains of length 2 of the second kind*</A> (2012).