Sequence of Composite Mersenne Numbers

Theorem
The sequence of Mersenne numbers which are composite begins:
 * $2047, 8 \, 388 \, 607, 536 \, 870 \, 911, 137 \, 438 \, 953 \, 471, 2 \, 199 \, 023 \, 255 \, 551,\ldots$

The sequence of corresponding indices $p$ such that $2^p - 1$ is composite begins:
 * $11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, \ldots$

The sequence of corresponding integers $n$ such that the $n$th prime number $p \left({n}\right)$ is such that $2^{p \left({n}\right)} - 1$ is composite begins:
 * $5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, \ldots$

Proof
Established by inspecting the sequence of Mersenne numbers:
 * $3, 7, 31, 127, 2047, 8191, 131 \, 071, 524 \, 287, 8 \, 388 \, 607, 536 \, 870 \, 911, 2 \, 147 \, 483 \, 647, \ldots$

and removing from it the sequence of Mersenne primes:
 * $3, 7, 31, 127, 8191, 131 \, 071, 524 \, 287, 2 \, 147 \, 483 \, 647, \ldots$