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 is a practical number 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.

Mathematics Subject Classification (2010):

• Primary: 11B65
• Secondary: 11N25

Key words:

• Central binomial coefficient
• practical number