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# Some properties of Zumkeller numbers and k-layered numbers

 Mahanta, Pankaj, Saikia, Manjil and Yaqubi, Daniel 2020. Some properties of Zumkeller numbers and k-layered numbers. Journal of Number Theory 217 (12) , pp. 218-236. 10.1016/j.jnt.2020.05.003 Item availability restricted.

## 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.

Item Type: Article Publication Published Mathematics Q Science > QA Mathematics Elsevier 0022-314X The Leverhulme Trust 1 September 2020 20 May 2020 25 Nov 2020 04:41 http://orca.cf.ac.uk/id/eprint/134550

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