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.

中文翻译：

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实用的中心二项式系数

摘要

一个实际的数字是一个正整数n ，这样所有小于n 的正整数都可以写成n的不同除数之和。Leonetti 和 Sanna 证明，当x → + ∞ 时，中心二项式系数是适用于所有正整数n ≤ x但最多为O ( x ^0.88097 ) 例外的数字。我们通过减少 exp ( C (log x ) ^4/5 log log x )的异常数量来改进这个结果，其中C > 0 是一个常数。