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Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)r = λ(n)s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.

Values of the Euler Function in Various Sequences