Let $n \in \mathbb{N}$ be given. What is the maximal value of the least common multiple of $x_1, \ldots, x_k$, all positive integers, when
$$x_1 + \ldots + x_k = n \: ,$$ and what is the asymptotic value of $\max( \mathrm{LCM}( x_1, \ldots, x_k))$?
Here's what I've found: We need to set $x_1, \ldots , x_k$ to as many prime numbers as possible. So if, say, $n=5$, we set $5 = 2 + 3 $ with $\mathrm{LCM}(2,3) =6$, and if $n=10$, we set $10 = 2+3+5$, $\mathrm{LCM}(2,3,5)=30$.
The sum of all prime numbers less than $x$ are
$$\sum_{p \leq x} p \approx \mathrm{Li} (x^2) ,$$ (see What is the sum of the prime numbers up to a prime number $n$? , where Li is the logarithmic integral), and the product of prime numbers less than $x$ is given by the primorial function:
$$\prod_{p \leq x } p \approx \exp{x}$$ (see https://en.wikipedia.org/wiki/Primorial#Definition_for_natural_numbers ). Combining these formulas, I get
$$\max( \mathrm{LCM}( x_1, \ldots, x_k) \approx \exp \left( \sqrt{ \mathrm{Li}^{-1} ( n) } \right) $$ but is this correct?
