For a positive integer $n$, let $f(n)$ be the maximum value of $\mathrm{LCM}(S)$ among multisets $S$ of positive integers satisfying $\sum_{i \in S} (i-1) = n$.

What is known about upper bounds for $f(n)$?

The best that I can do is $\log f(n) \leq \sqrt{n/2} \log (4n/3)$ for $n \geq 6$. This comes by looking at the product of distinct elements of $S$, which is certainly greater than $\mathrm{LCM}(S)$: Let $S$ have $k$ distinct elements. Then $n \geq k(k-1)/2$ so $k \leq \sqrt{3n}$. By the AM-GM inequality, $f(n) \leq ((n+k)/k)^k \leq (2n/k)^k$. Now $k(\log (2n) - \log k)$ is increasing for $k < 2n/e$, and $\sqrt{3n} < 2n/e$ for $n \geq 6$. So $f(n) \leq (2n/\sqrt{3n})^{\sqrt{2n}}$.

The above argument ignores the fact that distinct elements of $S$ may have common factors, so it may be possible to do better than this.

### Motivation

Let $G$ be a semisimple algebraic group over $\mathbb{Q}$. Let $\rho$ be an irreducible representation of $G$ such that $D = \mathrm{End} \rho$ is a central division algebra over $\mathbb{Q}$, with $\dim D = m^2$. In other words, $\rho \otimes \bar{\mathbb{Q}}$ has one irreducible component with multiplicity $m$.

I am interested in how large $m$ can be, given that $G$ has rank $n$.

By Tits' paper in J. Reine Angew. Math 247, $D$ is in the image of a certain homomorphism $\Lambda/\Lambda_0 \to \mathrm{Br} \mathbb{Q}$, where $\Lambda$ is the group of characters of a maximal torus and $\Lambda_0$ is the subgroup of $\Lambda$ generated by roots.

Since the base field is $\mathbb{Q}$, $m$ is the same as the order of $D$ in $\mathrm{Br}\mathbb{Q}$, so $m$ is less than or equal to the exponent of $\Lambda/\Lambda_0$.

Suppose that all simple components of the root system of $G$ have type A, and let $S$ be the multiset of ranks of simple components of $G$. Then $\Lambda/\Lambda_0$ is a subgroup of $\prod_{r \in S} \mathbb{Z}/(r+1)\mathbb{Z}$. (Components of types other than A contribute factors to $\Lambda/\Lambda_0$ with exponent at most 12, so can be ignored.) So the exponent of $\Lambda/\Lambda_0$ is at most $f(n)$ as defined above.