On Some Problems of Makowski-Schinzel and Erdos Concerning the Arithmetical Functions phi and sigma

Let sigma (n) denote the sum of the positive divisors of the integer n, and let phi denote Euler's function, that is, phi (n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Makowski and Schinzel that sigma ( phi (n))/n >= 1/2 for all n. We show that sigma ( phi (n))/n -> inf on a set of numbers n of asymptotic density 1. In addition, we study the average order of sigma ( phi (n))/n as well as its range. We use similar methods to prove a conjecture of Erdos that phi (n - phi (n))