Find the maximum value of $\text{lcm}(n_1,n_2,...,n_k)$ given $n_1+...+n_k=X\in\mathbb{Z^+}$,where $n_1,...,n_k$ and $k$ are to be determined.
I came across this as a lemma while solving a group theory problem.
By AMGM, $$\frac{n_1+...+n_k}{k}\ge \sqrt{n_1...n_k}\\\iff \frac{X}{k}\ge\sqrt{\text{LCM$\times$ GCD}}\\\iff\text{LCM}\le\frac{X^2}{k^2\times{\text{GCD}}}$$
Now the LCM is obtained if and only if all the $n$ are equal, but this becomes tricky. If we choose this, then the LHS bound changes, so perhaps we could do better if we chose a better bound but didn't get equality. Although this AMGM result looks huge, I would not be surprised if it was useless.
I've seen a similar problem before which is "Find the maximum value of $n_1n_2...n_k$ such that $n_1+...+n_k=100$". This is solved by taking as many threes as we can, then the rest twos. If in our problem we chose only primes, then the two problems are the same, so following the same principle, perhaps it is best to choose primes in ascending order ($n_1=2,n_2=3,n_3=5...$) until we get as close to $X$ as we can, then whatever is left over doesn't matter because we used up the primes anyway(so it won't increase the lcm).
Thanks in advance.
