Central binomial coefficients divisible by or coprime to their indices

Let A be the set of all positive integers n such that n divides the central binomial coefficient (2nn). Pomerance proved that the upper density of A is at most 1−log2. We improve this bound to 1−log2−0.05551. Moreover, let B be the set of all positive integers n such that n and (2nn) are relatively prime. We show that #(B∩[1,x])≪x/logx−−−−√ for all x>1