On some problems of M/akowski–Schinzel and Erdős concerning the arithmetical functions $\phi $ and $\sigma $
Let $\sigma (n)$ denote the sum of 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 Mąkowski and Schinzel that $\sigma (\phi (n))/n\ge 1/2$ for all $n$. We show that $\sigma (\phi (n))/n\to \infty $ 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 Erdős that $\phi (n-\phi (n))<\phi (n)$ on a set of asymptotic density 1.