Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$?

$45$ satisfies the Erdős 1142 property: $45 - 2 = 43$, $45 - 4 = 41$, $45 - 8 = 37$, $45 - 16 = 29$, $45 - 32 = 13$.