Do we know a $2^{M_i}-1$ that's not prime?

Question

Do we know a $2^{M_i}-1$ that's composite, i.e. not prime, where $M_i$ is a Mersenne prime number of the form $2^p-1$?

For example

$2^7-1=127$ is not an example because 127 is actually prime.
$2^{11}-1 = 2047 = 23*89$ is not an example because $11$ is not a Mersenne prime ($2^p-1$).

$2^{8191}-1$ is composite, though $8191=M_{13}$ is a Mersenne prime. — lulu, Nov 26 '18 at 13:18
here is the sequence of Mersenne exponents. here is the sequence of Mersenne primes. — lulu, Nov 26 '18 at 13:20

score 2 · Accepted Answer · answered Nov 26 '18 at 14:35

As remarked in the comments: $M_{13}=8191$ is the smallest Mersenne Prime which is not also a Mersenne exponent.

Of course, the size of the numbers involved makes it difficult to compute much directly. Still, the available lists are adequate to the purpose at hand. The Mersenne primes are sequence A000668 on OEIS, and the Mersenne exponents are A000043.

Do we know a $2^{M_i}-1$ that's not prime?

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