Zumkeller数和$k$ -分层数的一些性质
Some properties of Zumkeller numbers and $k$-layered numbers
摘要：
完满数的概念是由Zumkeller在2003年提出的Zumkeller或整数完满数。正整数$n$是一个Zumkeller数，如果它的除数可以被分成两个具有相同和的集合，这将是$σ(n)/2$。进一步推广，我们称$n$为一个$k$ -分层数，如果它的因子可以被划分成具有相等和的$k$集合。本文完全刻画了具有两个不同素因子的Zumkeller数，并给出了具有两个以上不同素因子的Zumkeller数的素因子分解的界.我们还刻画了具有两个不同素因子的$k$ -分层数和具有两个以上不同奇素因子的偶$k$ -分层数。本文还讨论了与这些数有关的其它结果及其与实际数和调和平均数的关系。
英文摘要：
Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be $σ(n)/2$. Generalizing even further, we call $n$ a $k$-layered number if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we completely characterize Zumkeller numbers with two distinct prime factors and give some bounds for prime factorization in case of Zumkeller numbers with more than two distinct prime factors. We also characterize $k$-layered numbers with two distinct prime factors and even $k$-layered numbers with more than two distinct odd prime factors. Some other results concerning these numbers and their relationship with practical numbers and Harmonic mean numbers are also discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
