Abstract

Let n > 2 be a positive integer and let ϕ denote Euler’s totient function. Define ϕ^{1}(n) = ϕ(n) and ϕ^{k}(n) = ϕ(ϕ^{k−1}(n)) for all integers k ≥ 2. Define the arithmetic function S by S(n) = ϕ(n) + ϕ^{2}(n) + ⋯ + ϕ^{c}(n) + 1, where ϕ^{c}(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.