Practical central binomial coefficients

Abstract

A practical number is a positive integer $n$ such that all positive integers less than $n$ can be written as a sum of distinct divisors of $n$. Leonetti and Sanna proved that, as $x\to +\infty$, the central binomial coefficient $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ is a practical number for all positive integers $n\le x$ but at most $O(x^{0.88097})$ exceptions. We improve this result by reducing the number of exceptions to $\exp (C(\log x{)}^{4/5}\log \log x)$, where $C>0$ is a constant.