$\DeclareMathOperator\lcm{lcm}$We know that $\operatorname{lcm}(1,\dotsc,n)$ is approximately $e^n$ and we also know that $\gcd(2^a-1, 2^b-1)=2^{\gcd(a,b)}-1$.

I wonder if there exists an upper bound/lower bound/approximation for $\operatorname{lcm}(2^1-1, 2^2-1,\dotsc,2^n-1)$.