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.
Citation: Shichun Yang, Qunying Liao, Shan Du, Huili Wang. The explicit formula and parity for some generalized Euler functions[J]. AIMS Mathematics, 2024, 9(5): 12458-12478. doi: 10.3934/math.2024609
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.
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Shichun Yang, Qunying Liao, Shan Du, Huili Wang. The explicit formula and parity for some generalized Euler functions[J]. AIMS Mathematics, 2024, 9(5): 12458-12478. doi: 10.3934/math.2024609
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