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This is the "Cunningham chain of the second kind of length 2", here is the OEIS reference and here is the Wikipedia entry; record primes of this type are listed here, 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 $1.32\, N/\log^2 N$.