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>; a few years ago the largest known prime $p$ such that $2p-1$ is also prime was $p=648309*2^{148310}+1$, perhaps now a larger $p$ is known.