Amdeberhan-Medina-Moll Arctangent Sum Conjecture

Let $x_n = \tan\left(\sum_{k=1}^{n} \arctan k\right)$. Amdeberhan, Medina and Moll conjectured that $x_n \notin \mathbb{Z}$ for every $n \ge 5$. Any integer value $x_n = m$ must satisfy $|m| \ge e^{(1/2+o(1)) n \log n}$, which forces $\#\{1 \le n \le N : x_n \in \mathbb{Z}\} = O(\log N)$. The conjecture therefore holds for a density-one set of $n$, improving on the previously known density of $120/817 \approx 0.147$.