On Zumkeller numbers

被引：11

作者：

Peng, Yuejian ^{[1
]}

Rao, K. P. S. Bhaskara ^{[2
]}

机构：

[1] Hunan Univ, Sch Math, Changsha 410082, Hunan, Peoples R China

[2] Indiana Univ NW, Dept Comp Informat Syst, Gary, IN 46408 USA

来源：

JOURNAL OF NUMBER THEORY | 2013年 / 133卷 / 04期

基金：

中国国家自然科学基金;

关键词：

Perfect numbers; Practical numbers; Zumkeller numbers;

D O I：

10.1016/j.jnt.2012.09.020

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Text. A positive integer n is perfect if the sum of the proper positive divisors of n equals n. Generalizing this we call n a Zumkeller number if the set of its positive divisors can be partitioned into two disjoint subsets of equal sum. Similarly we call n a half-Zumkeller number if the set of its proper positive divisors can be so partitioned. A study of Zumkeller numbers, half-Zumkeller numbers and their relation to practical numbers is undertaken in this paper. Clark et al. (2008) [1] announced some results about Zumkeller numbers and half-Zumkeller numbers, and suggested two conjectures. In the present paper we shall settle one of the conjectures, prove the second conjecture in some special cases, and prove several results related to the second conjecture. We shall also show that if there is an even Zumkeller number that is not half-Zumkeller it is bigger than 7233 498 900. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=z85qyvIorBE. (C) 2012 Elsevier Inc. All rights reserved.

引用

页码：1135 / 1155

页数：21