Some properties of Zumkeller numbers and $k$-layered numbers

TL;DR

This paper explores properties of Zumkeller and $k$-layered numbers, providing characterizations, bounds, and relationships with other number classes, expanding understanding of their divisor partitioning structures.

Contribution

It offers a complete characterization of Zumkeller numbers with two prime factors and bounds for those with more, also analyzing $k$-layered numbers with specific prime factor conditions.

Findings

Characterization of Zumkeller numbers with two prime factors

Bounds for prime factors in complex Zumkeller numbers

Analysis of $k$-layered numbers with specific prime factorization

Abstract

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 $\sigma(n)/2$. 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.