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

@article{Ford2019DivisibilityOT,
  title={Divisibility of the central binomial coefficient \$\binom \{2n\}\{n\}\$},
  author={Kevin Ford and Sergei Konyagin},
  journal={arXiv: Number Theory},
  year={2019}
}
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. 

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