The Solvability Of Two Equations Related To The Generalized Euler Function

Let n=p1?1p2?2…pk?k be a positive integer,where pi are distinct primes and?i 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 ?(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 ?e(n)=2t?(n)and ?e(n)=pt?(n),and then determines the relevant solutions under given conditions.