On unitary Zumkeller numbers

Bhabesh Das
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 436–442
DOI: 10.7546/nntdm.2024.30.2.436-442
Full paper (PDF, 428 Kb)

Details

Authors and affiliations

Bhabesh Das
Department of Mathematics, Arya Vidyapeeth College (Autonomous)
Guwahati – 16, Assam, India

Abstract

It is well known that if n is a Zumkeller number, then the positive divisors of n can be partitioned into two disjoint subsets of equal sum. Similarly for unitary Zumkeller number n, the unitary divisors of n can be partitioned into two disjoint subsets of equal sum. In this article, we have derived some results related to unitary Zumkeller number, unitary half-Zumkeller number and also presented some numerical examples.

Keywords

• Zumkeller number
• Unitary Zumkeller number
• Divisor function
• Unitary divisor function

2020 Mathematics Subject Classification

• 11A25

References

1. Clark, S., Dalzell, J., Holliday, J., Leach, D., Liatti, M., & Walsh, M. (2008). Zumkeller numbers. In: Mathematical Abundance Conference, Illinois State University, April 18, 2008.
2. Great Internet Mersenne Prime Search (2024). Available online at: https://www.mersenne.org
3. Peng, Y., & Bhaskara Rao, K. P. S. (2013). On Zumkeller numbers. Journal of Number Theory, 133(4), 1135–1155.
4. Subbarao, M. V., & Warren, L. J. (1966). Unitary perfect numbers. Canadian Mathematical Bulletin, 9(2), 147–153.
5. The Online Encyclopedia of Integer Sequences (2010). Zumkeller or integer-perfect numbers: Numbers n whose divisors can be partitioned into two disjoint sets with equal sum. Available online at: https://oeis.org/A083207 .
6. The Online Encyclopedia of Integer Sequences (2010). Unitary Zumkeller numbers: Numbers k whose unitary divisors can be partitioned into two disjoint subsets whose sums are both usigma(k)/2. Available online at: https://oeis.org/A290466 .
7. The Online Encyclopedia of Integer Sequences (2010). Unitary half-Zumkeller numbers: Numbers k whose unitary proper divisors can be partitioned into two disjoint sets whose sums are equal. Available online at: https://oeis.org/A290467 .

Manuscript history

• Received: 31 March 2023
• Revised: 31 May 2024
• Accepted: 12 July 2024
• Online First: 16 July 2024

Copyright information

Ⓒ 2024 by the Author.
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).