Let A be the set of all positive integers n such that n divides the central binomial coefficient (2n, n). Pomerance proved that the upper density of A is at most 1 - log 2. We improve this bound to 1 - log 2 - 0.05551. Moreover, let B be the set of all positive integers n such that n and (2n, n) are relatively prime. We show that #(B ^ [1, x]) << x / sqrt(log x) for all x > 1.
Central binomial coefficients divisible by or coprime to their indices
Sanna, Carlo
2017-01-01
Abstract
Let A be the set of all positive integers n such that n divides the central binomial coefficient (2n, n). Pomerance proved that the upper density of A is at most 1 - log 2. We improve this bound to 1 - log 2 - 0.05551. Moreover, let B be the set of all positive integers n such that n and (2n, n) are relatively prime. We show that #(B ^ [1, x]) << x / sqrt(log x) for all x > 1.
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