与约数和函数σ(n)有关的一些不等式的解
The Solutions of Some Inequalities of the Sum of Distinct Divisors Function

作者: 吴 莉 * , 杨仕椿 :阿坝师范高等专科学校数学系;

关键词: 约数和函数正整数解标准分解式 Sum of Divisors Positive Integer Solution Standard Decomposition

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.

文章引用: 吴 莉 , 杨仕椿 (2013) 与约数和函数σ(n)有关的一些不等式的解。 理论数学, 3, 107-111. doi: 10.12677/PM.2013.32017

参考文献

[1] R. K. Guy. Unsolved problems in number theory. New York: Springer-Verlag, 1981: 25-56.

[2] 华罗庚. 数论导引[M]. 北京: 科学出版社, 1979: 13-14.

[3] P. Erdos. Remarks on number theory II: Some problems on the σ function. Acta Arithmetica, 1959, 5: 171-177.

[4] A. Makowski, M. A. Schinzel. On the functions σ(n) and φ(n). Colloquium Mathematicum, 1964, 113: 95-99.

[5] 柯召, 孙琦. 论一类型积性数论函数方程[J]. 四川大学学报(自然科学版), 1965, 2(1): 1-10.

[6] G. L. Cohn. On a conjecture of Makowski and Schinzel. Colloquium Mathematicum, 1994, 74: 1-8.

[7] D. F. Luca, C. Pomerance. On some problems of Makowski-Schinzel and Erdos concerning the arithmetical functions φ and σ. Colloquium Mathematicum, 2002, 92(1): 111-130.

[8] A. Makowski, A. Schinzel. On the functions φ(n) and σ(n). Colloquium Mathematicum, 1965, 13(1): 95-99.

[9] C. Pomerance. On the composition of the arithmetic functions φ and σ. Colloquium Mathematicum, 1989, 58: 11-15.

[10] M. Bencze. Open question 2327. Octogon Mathematical Magazine, 2006, 14(2): 872.

[11] M. Bencze. Proposed problem 4935. Octogon Mathematical Magazine, 2004, 12(2B): 824.

[12] 乐茂华. 关于数论函数δ(n)的一个公开问题[J]. 广东教育学院学报, 2007, 27(5): 9-10.

[13] 乐茂华. 关于数论函数δ(n)的一个问题[J]. 周口师范学院学报, 2007, 27(5): 1-2.

分享
Top