Values of the Euler Function in Various Sequences

This is a preprint of an article published in Monatshefte für Mathematik (2005), Volume 146, Number 1, 1-19, DOI: 10.1007/s00605-005-0302-7. The final publication is available at springerlink.com.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 deg f > 1