A note on generalized Zumkeller numbers

Jayanta Kalita and Helen K. Saikia
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 150–161
DOI: 10.7546/nntdm.2026.32.1.150-161
Full paper (PDF, 254 Kb)

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Authors and affiliations

Jayanta Kalita
Department of Mathematics, Gauhati University
Guwahati, Assam, India

Helen K. Saikia
Department of Mathematics, Gauhati University
Guwahati, Assam, India

Abstract

A positive integer n is called an e-Zumkeller number if the exponential divisors of n can be partitioned into two disjoint subsets of equal sum. Generalizing the concept of e-Zumkeller numbers, we define multiplicatively e-Zumkeller numbers. In addition, generalizing the concepts of s-Zumkeller numbers and m-Zumkeller numbers, we define two new variants of Zumkeller numbers called (+s)-Zumkeller numbers and (+m)-Zumkeller numbers, considering even positive divisors. We present some examples in support of these two types of positive integers and study their characteristics.

Keywords

• Perfect number
• Zumkeller number
• m-Zumkeller number
• s-Zumkeller number

2020 Mathematics Subject Classification

• 11A25

References

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Manuscript history

• Received: 10 October 2025
• Revised: 18 February 2026
• Accepted: 28 February 2026
• Online First: 6 March 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).