The computing formula for generalized Euler functions

Let n and e be positive integers. Based on elementary methods and techniques, the explicit formula for φe(n)(e=pr, ∏ti=1qi) was given for the case q1≡…≡qt≡1(mod p) or q1≡…≡qt≡-1(mod p), where p,q1,…,qt are distinct primes, t and r are both positive integers, thus generalizing the previous results.