Practical central binomial coefficients

Abstract

A practical number is a positive integer n such that all positive integers less than n can be written as a sum of distinct divisors of n. Leonetti and Sanna proved that, as x → +∞, the central binomial coefficient (2n/n) is a practical number n for all positive integers n ≤ x but at most O(x^0.88097) exceptions. We improve this result by reducing the number of exceptions to exp(C(log x)^4/5 log log x), where C > 0 is a constant.