On a new additive arithmetic function related to a fixed integer

Mihoub Bouderbala and Meselem Karras
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 575–580
DOI: 10.7546/nntdm.2022.28.3.575-580
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Authors and affiliations

Mihoub Bouderbala
Department of Mathematics, University of Djilali Bounaama
FIMA Laboratory, Ain Defla, Khemis Miliana, Algeria

Meselem Karras
Department of Mathematics, University of Djilali Bounaama
FIMA Laboratory, Ain Defla, Khemis Miliana, Algeria

Abstract

The main purpose of this paper is to define a new additive arithmetic function related to a fixed integer k\geq 1 and to study some of its properties. This function is given by

    \begin{equation*} \ f_{k}\left( 1\right) =0\text{ and }f_{k}\left( n\right) =\sum_{p^{\alpha}\parallel n}\left( k,\alpha \right) , \end{equation*}

such that (a, b) denotes the greatest common divisor of the integers a and b.

Keywords

  • Arithmetic function
  • Greatest common divisor

2020 Mathematics Subject Classification

  • 11N37
  • 11A25

References

  1. Atanassov, K. (1987). New integer functions, related to φ and σ functions. Bulletin of Number Theory and Related Topics, XI(1), 3–26.
  2. Atanassov, K., & Sándor, J. (2018). On a new arithmetic function. Notes on Number Theory and Discrete Mathematics, 24(1), 3–10.
  3. Bagdasar, O., & Tatt, R. (2018). On some new arithmetic functions involving prime divisors and perfect powers. Electronic Notes in Discrete Mathematics, 70, 9–15.
  4. Panaitopol, L. (2004). Properties of the Atanassov functions. Advanced Studies on Contemporary Mathematics, 8(1), 55–99.
  5. Shubin, A. V. (2017). Asymptotic behavior of functions Ω(k, n) and ω(k, n) related to the number of prime divisors. Diskretnaya Matematika, 29(3), 133–143.

Manuscript history

  • Received: 2 April 2022
  • Revised: 20 September 2022
  • Accepted: 21 September 2022
  • Online First: 27 September 2022

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Cite this paper

Bouderbala, M., & Karras, M. (2022). On a new additive arithmetic function related to a fixed integer. Notes on Number Theory and Discrete Mathematics, 28(3), 575-580, DOI: 10.7546/nntdm.2022.28.3.575-580.

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