Erdos, Graham, Ruzsa, and Straus proved that if
$$
f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p}
$$
then there is some constant $c < 1$ such that for all large $n$
$$
f(n) \leq c \log\log n.
$$
[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., _On the prime factors of $\binom{2n}{n}$_. Math. Comp. (1975), 83-92.
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