Let define $\xi$ an arithmetic fonction such that : $\xi(n)=\underset{k_1+...+k_l=n} \max lcm (k_1,...k_l)$, where $k_1,...,k_l\in \mathbb{N}^{*}$.
I have to prove that $\xi(n)=\underset{p_1^{a_1}+...+p_s^{a_s}\le n}\max (p_1^{a_1}...p_s^{a_s})$, where $p_1,...,p_s$ are distinct prime numbers.
I meet difficulties to write it correctly. Here is my attempt.
Using the fundamental theorem of arithmetic I have :
$k_1=p_{1,1}^{a_{1,1}}...p_{s,1}^{a_{s,1}}, \ k_2=p_{1,2}^{a_{1,2}}...p_{s,2}^{a_{s,2}}, \ \ ..., \ \ k_l=p_{1,l}^{a_{1,l}}...p_{s,l}^{a_{s,l}}$.
Then by a strong formula $lcm(k_1,...,k_l)=\prod \limits_{i,k}p_{i,k}^{\max(a_{i,k})}$ but here I don't know how to write it well too many indexes...
Maybe I can say that $p_1$ is the first prime number which is in every $k_i$ but I'm quickly confused.
Also if we have $k_1+...+k_l=n$ then $p_1^{a_1}+...+p_s^{a_s}\le n.$
Thanks in advance !
NB : Apparently this function is well-known and here is an asymptotic result about it : $\lim_{n \to +\infty} \frac{\ln(\xi(n))}{\sqrt{(n\ln(n))}}=1$.