Erdős problem 731

Let $A(n)$ be the least positive integer not dividing $\binom{2n}{n}$. Erdos asked for the behaviour of $A(n)$ for reasonable $n$. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals against the scale $F_X = \sqrt{2}(\log 2)^{1/4} L^{1/4} \exp\sqrt{(\log 2)L}$ with $L = \log(2X)$.