New properties of arithmetic functions related to gcd and lcm

Brahim Mittou
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 91–97
DOI: 10.7546/nntdm.2025.31.1.91-97
Full paper (PDF, 219 Kb)

Details

Authors and affiliations

Brahim Mittou
Department of Mathematics, University Kasdi Merbah, Ouargla
EDPNL & HM Laboratory, ENS of Kouba, Algiers, Algeria

Abstract

This paper explores additional properties of the arithmetic functions f_\alpha(n) and g_\alpha(n), defined respectively by f_\alpha(n) = \prod_{i=1}^r p_i^{(e_i, \alpha)} and g_\alpha(n) = \prod_{i=1}^r p_i^{[e_i, \alpha]}, where n = \prod_{i=1}^r p_i^{e_i} is the prime factorization of a positive integer n>1, (a, b) and [a, b] denote, respectively the greatest common divisor and the least common multiple of any two integers a and b. These functions and some of their properties have been introduced and investigated in previous works. In this paper, we establish several new theorems that reveal deeper insights into the relationships between these functions.

Keywords

  • Arithmetic function
  • Greatest common divisor
  • Least common multiple

2020 Mathematics Subject Classification

  • 11A25

References

  1. Apostol, T. M. (1976). Introduction to Analytic Number Theory. Springer-Verlag, New York.
  2. Mittou, B. (2022). New properties of an arithmetic function. Mathematica Montisnigri, 53, 5–11.
  3. Mittou, B. (2023). New arithmetic function related to the least common multiple. Journal of Science and Arts, 23(1), 123–128.
  4. Mittou, B., & Derbal, A. (2021). On new arithmetic function relative to a fixed positive integer. Part 1. Notes on Number Theory and Discrete Mathematics, 27(1), 22–26.
  5. Sándor, J., & Atanassov, K. (2021). Arithmetic Functions. Nova Science, New York.
  6. Subbarao, M. V. (1972). On some arithmetic convolutions. In: The Theory of Arithmetic Functions, LNM 251, pp. 247–271, Springer-Verlag, New York.

Manuscript history

  • Received: 19 November 2024
  • Revised: 8 April 2025
  • Accepted: 9 April 2025
  • Online First: 9 April 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).

Related papers

Cite this paper

Mittou, B. (2025). New properties of arithmetic functions related to gcd and lcm. Notes on Number Theory and Discrete Mathematics, 31(1), 91-97, DOI: 10.7546/nntdm.2025.31.1.91-97.

Comments are closed.