The Explicit Formula and Parity for Some Generalized Euler Functions

TL;DR

This paper derives explicit formulas for generalized Euler functions with e=8,12, and establishes conditions for their oddness using elementary methods.

Contribution

It provides new explicit formulas and parity conditions for generalized Euler functions for specific values of e, expanding understanding of their properties.

Findings

Explicit formulas for (n) with e=8,12

Necessary and sufficient conditions for (n) to be odd

Elementary methods used for derivation

Abstract

Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is obtained, respectively.