Furmat

Suppose such an integer $n \gt 1$ exists, and let $p$ be the least prime factor of $n.$ Now since $2^{n} - 1$ is odd we know that $p \gt 2.$

Now since $n | 2^{n} - 1$ we must also have that $p | 2^{n} - 1.$

Also, by Fermat's Little Theorem, we know that $2^{p - 1} \equiv 1 \pmod{p} \Longrightarrow p | 2^{p - 1} - 1.$

This implies that $p|2^{d} - 1$ where $d = gcd(n, p - 1).$ But since $p$ is the least prime factor of $n$ the only possible value for $d$ is $1,$ which would imply that $p|1,$ which is absurd. Thus the original supposition that such an integer $n$ exists must be false, i.e., there are $\boxed{0}$ such integers $n.$