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Research Article | Open Access

The explicit formula and parity for some generalized Euler functions

Shichun Yang1,2( ), Qunying Liao2, Shan Du2, Huili Wang2
School of Mathematics, ABa Teachers University, Wenchuan 623000, Sichuan, China
School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, Sichuan, China
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Abstract

Utilizing elementary methods and techniques, the explicit formula for the generalized Euler function φ e ( n ) ( e = 8 , 12 ) has been developed. Additionally, a sufficient and necessary condition for φ 8 ( n ) or φ 12 ( n ) to be odd has been obtained, respectively.

Keywords

generalized Euler functionexplicit formulaMöbius functionparity
CLC number: 11A25, 11B34, 11B65

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AIMS Mathematics
Volume 9 Issue 5,
May 2024
Pages 12458-12478
DOI: 10.3934/math.2024609

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Cite this article:
Yang S, Liao Q, Du S, et al. The explicit formula and parity for some generalized Euler functions. AIMS Mathematics, 2024, 9(5): 12458-12478. https://doi.org/10.3934/math.2024609

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Received: 18 December 2023
Revised: 11 March 2024
Accepted: 21 March 2024
Published: 15 May 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)