# Practical numbers among the binomial coefficients

## TL;DR

This paper demonstrates that most binomial coefficients are practical numbers, providing bounds on the number of exceptions, and shows that central binomial coefficients are practical for most integers, with some open questions posed.

## Contribution

It establishes that the majority of binomial coefficients are practical numbers and quantifies the rarity of exceptions, advancing understanding of their distribution.

## Key findings

- Most binomial coefficients are practical numbers.
- The number of non-practical binomial coefficients grows slower than n.
- Central binomial coefficients are practical for most integers with few exceptions.

## 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.