$\DeclareMathOperator\lcm{lcm}$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 $\operatorname{lcm}(2^1-1, 2^2-1,\dotsc,2^n-1)$.



Do we know any bound on $\operatorname{lcm}(2^1-1, 2^2-1,\dots,2^n-1)$?
Here my guess that seems to be confirmed by numerical computation.


Let $(u\_n)\_{n \geq 0}$ be a non-degenerate Lucas sequence of the first kind with a dominant positive real root $\alpha$.
In other words, $u\_0 = 0$, $u\_1 = 1$, $u\_{n+2} = a u\_{n+1} + b u\_n$ for each $n \geq 0$ for some relatively prime integers $a, b$; and the roots $\alpha, \beta$ of the characteristic polynomial $X^2 - aX - b$ are real and satisfy $\alpha > |\beta|$.


Can we find some asymptotic for $L\_n := \text{lcm}\{u\_k : k \leq n\}$ as $n \to \infty$?


Note that your case of $u\_k = 2^k-1$ is given by $a = 3$, $b = -2$ (and hence $\alpha = 2$).


It holds the product formula 
$$u\_n = \prod\_{\substack{d \,\mid\, n \\ d > 1}} \Phi\_d ,$$
where 
$$\Phi\_d := \prod\_{\zeta \text{ primitive $d$-th root of unity}} (\alpha - \zeta \beta)$$
is the homogeneous cyclotomic polynomial of $\alpha$ and $\beta$.
Thus, clearly $L\_n \mid \prod\_{k \leq n} \Phi\_k$. On the other hand, it is not true that the $\Phi\_d$'s are pairwise coprime, however if $\gcd(\Phi\_k, \Phi\_h) > 1$ for some $k < h$, then $\gcd(\Phi\_k, \Phi\_h)$ is a prime number. Therefore, it seems quite possible that $L\_n$ is not too smaller than $\prod\_{k \leq n} \Phi\_k$, so I guess $\log(L\_n) \approx \sum\_{k \leq n} \log(\Phi\_k)$. Since $\alpha$ is dominant and $\Phi\_k$ has degree $\varphi(k)$, it should be $\Phi\_k \approx \alpha^{\varphi(k)}$. Finally, from the asymptotic formula
$$\sum\_{k \leq n} \varphi(k) = \frac{3}{\pi^2}n^2 + O(n)$$
my conjecture is that
$$\log(L\_n) \sim \frac{3\log\alpha}{\pi^2} \cdot n^2 ,$$
as $n \to \infty$.


**P.S.** My conjecture is true. The question was studied in much generality by S. Akiyama and F. Luca [1]. In particular, their Theorem 2 gives the guessed asymptotic for $L\_n$, also with a good error term.


[1] S. Akiyama and F. Luca, On the least common multiple of Lucas subsequences, Acta Arith. 161 (2013), 327--349


