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:
![](https://image.hanspub.org/IMAGE/Edit_2e1cdb14-1f73-48cd-9794-f31141907870.jpg)
,
![](https://image.hanspub.org/IMAGE/Edit_3278e6fd-8c19-40a8-b049-9ad551cb826f.jpg)
,
![](https://image.hanspub.org/IMAGE/Edit_8d64a187-fd41-4bf3-8bcb-5de9b649b6ac.jpg)
,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:
![](https://image.hanspub.org/IMAGE/Edit_1ab47c1b-0c1a-4ca6-8339-5d04b6fac3f6.jpg)
,
![](https://image.hanspub.org/IMAGE/Edit_5857653e-e183-47d1-b2b0-486f96d34aef.jpg)
and
![](https://image.hanspub.org/IMAGE/Edit_66df86f2-e9a4-46a0-82bd-22b8e545c502.jpg)
, where denotes the sum of distinct divisors of n.