Divisibility of the central binomial coefficient $\binom{2n}{n}$

TL;DR

This paper investigates the divisibility properties of central binomial coefficients, establishing positive densities for certain divisibility conditions and introducing a novel method to analyze large prime factors in sequences.

Contribution

It provides new results on the asymptotic density of integers dividing central binomial coefficients and introduces a method to analyze large prime factors in sequences.

Findings

Positive asymptotic density $c_\ell$ for integers $n$ with $n^\ell$ dividing $inom{2n}{n}$

Asymptotic formula for $c_\ell$ as $\ell \to \infty$

Estimate of the count of $n \le x$ coprime to $inom{2n}{n}$ as $cx/\log x$

Abstract

We show that for every fixed $\ell\in\mathbb{N}$, the set of $n$ with $n^\ell|\binom{2n}{n}$ has a positive asymptotic density $c_\ell$, and we give an asymptotic formula for $c_\ell$ as $\ell\to \infty$. We also show that $\# \{n\le x, (n,\binom{2n}{n})=1 \} \sim cx/\log x$ for some constant $c$. One novelty is a method to capture the effect of large prime factors of integers in general sequences.