摘要:
约数和函数是一类基本而又重要的数论函数。本文推广了由Bencze提出的两个公开问题的结论，证明了对于任意给定的正整数k和非零整数b，均存在无穷多个正整数n，使得以下三个不等式同时成立：
，
，
，其中为任意正整数n的不同约数之和。
Abstract: The sum of distinct divisors is a basic and important arithmetical function. In this paper, we extend the conclusions of two open problems proposed by Bencze and prove that, for any given positive integers k and non-zero integers b, there exists infinitely many positive integers n such that the following three inequalities hold simultaneously:
,
and
, where denotes the sum of distinct divisors of n.