The explicit formula and parity for some generalized Euler functions

Abstract

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

Keywords

• generalized euler function
• explicit formula
• möbius function
• parity