The Solvability Of Two Equations Related To The Generalized Euler Function

Let n=p1α1p2α2…pkαk be a positive integer,where pi are distinct primes andαi are positive integer(i=1,2,…,k).Suppose that e ∈ {2,3,4,6},p is an odd prime,and t is a positive integer.Denote ω(n)as the number of different prime factors of n.By using elementary methods and properties for the generalized Euler function,we study the solvability of the equations φe(n)=2tω(n)and φe(n)=ptω(n),and then determines the relevant solutions under given conditions.