Let n > 2 be a positive integer and let φ denote Euler's totient function. Define φ1(n) = φ(n) and φk(n) = φ(φk-1(n)) for all integers k ≥ 2. Define the arithmetic function S by S(n) = φ(n) + φ2(n) +...+ φc(n) + 1, where φc(n) = 2. We say n is a perfect totient number if S(n) = n. We give a list of known perfect totient numbers, and we give sufficient conditions for the existence of further perfect totient numbers
