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 $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, with $0\le k\le n$, that are not practical numbers, we show that \begin{equation*} f(n) < n^{1 - (\log 2 - δ)/\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 $x\ge 3$ and $0<\gamma <\delta <\log 2$. Furthermore, we prove that 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 also pose some questions on this topic.

10 pages, no figures