A formula similar to this coined in by Enzo Creti is gained as follows :
Instead of concatenating the Mersenne numbers $M_n$ and $M_{n-1}$ , we do it in reverse. The formula for this sequence is $$f(n)=(2^{n-1}-1)\cdot 10^d+2^n-1$$ where $d$ is the number of digits in the decimal expansion of $2^n-1$
The first few $n$ giving a prime are
$$[2, 3, 5, 6, 7, 9, 14, 26, 39, 41, 42, 46, 65, 161]$$
Now, if we note the residues of the primes modulo $7$ , all residues are present already (except $0$ of course). A coincidence ?
Further primes occur for $n=342,662,959,1794,4211,8254$
