On ψ-amicable numbers and their generalizations

Stoyan Dimitrov
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 768–775
DOI: 10.7546/nntdm.2025.31.4.768-775
Full paper (PDF, 197 Kb)

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Authors and affiliations

Stoyan Dimitrov
1 Faculty of Applied Mathematics and Informatics, Technical University of Sofia
Blvd. St.Kliment Ohridski 8, Sofia 1000, Bulgaria
2 Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences
Acad. G. Bonchev Str. Bl. 105, Sofia 1113, Bulgaria

Abstract

In this article, we study the properties of ψ-amicable numbers. We prove that their asymptotic density relative to the positive integers is zero. We also propose generalizations of ψ-amicable numbers.

Keywords

  • Dedekind ψ(n) function
  • ψ-amicable numbers

2020 Mathematics Subject Classification

  • 11A25
  • 11D72

References

  1. Dickson, L. E. (1913). Amicable number triples. The American Mathematical Monthly, 20, 84–92.
  2. Erdős, P. (1934). On the density of the abundant numbers. Journal of the London Mathematical Society, 9, 278–282.
  3. Erdős, P. (1955). On amicable numbers. Publicationes Mathematicae Debrecen, 4, 108–111.
  4. Mason, T. E. (1921). On amicable numbers and their generalizations. The American Mathematical Monthly, 28, 195–200.
  5. Penney, D. E., Pomerance, C., Brown, K., & Eynden, C. V. (1996). Pseudo-aliquot Sequences. Solution to Problem 10323. The American Mathematical Monthly, 103, 697–698.
  6. Sloane, N. J. A. (2025). The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org.
  7. Yanney, B. F. (1923). Another definition of amicable numbers and some of their relations to Dickson’s amicables. The American Mathematical Monthly, 30, 311–315.

Manuscript history

  • Received: 30 June 2025
  • Revised: 27 September 2025
  • Accepted: 18 October 2025
  • Online First: 3 November 2025

Copyright information

Ⓒ 2025 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).

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Cite this paper

Dimitrov, S. (2025). On ψ-amicable numbers and their generalizations. Notes on Number Theory and Discrete Mathematics, 31(4), 768-775, DOI: 10.7546/nntdm.2025.31.4.768-775.

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