Practical numbers among the binomial coefficients

Abstract

A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting $f(n)$ denote the number of binomial coefficients $\binom{n}{k}$, with $0 \leq k \leq n$, that are not practical numbers, we show that \begin{equation*} f(n) < n^{1 - (\log 2 - \delta)/\log \log n} \end{equation*} for all integers $n \in [3, x]$, but at most $O_\gamma(x^{1 - (\delta - \gamma) / \log \log x})$ exceptions, for all $x \geq 3$ and $0 < \gamma < \delta < \log 2$. Furthermore, we prove that the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions. We also pose some questions on this topic.