Odd Values ​​of The Generalized Euler Function (?) _e (n)

Let Z+ be the set of all positive integers, Euler functionφ(n) defined on Z+ is an important function in number theory. 1, i.e., the number of positive integers not greater than n and prime to n. It is excessively useful in number theory. Since 1970's,φ(n) has played key role in RSA public-key cryp-tosystem.In order to generalize Lehmer's congruences from modulo prime squares to modulo integer squares, Cai [2] defined the following generalized Euler function, for each e≥1, where [x] is the greatest integer not greater than x, i.e.,φe(n) is the number of positive integers not greater than [n/e] and prime to n. It's easy to verify that whereμ(n) is Mobius function.When e=1,φe(n) is the Euler functionφ(n). In this note, we study the parity of the generalized Euler functionφe(n). For e=4, we give sufficient and necessary condition for bothφe(n) andφe(n+1) be odd.