Here we define those primes $p$ for which $\operatorname{ord}_p(2)=s$, where $s$ is the minimum of the set $S$ of all divisors $d\mid p-1$ such that $2^d-1\geq p$.
For example: for $p=7$, $s=3$, $7\mid 2^3-1$ thus $\operatorname{ord}_p(2)=s=3$ ($7$ is such a prime).
Questions: how many such primes are there? Are such primes interesting?
Thanks.