Noncototients and Nonaliquots

Abstract

Let $\phi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for which the equation $m=n-\phi(n)$ has no solution. We also give a lower bound for the number of $m\le x$ for which the equation $m=\sigma(n)-n$ has no solution. Finally, we show the set of positive integers $m$ not of the form $(p-1)/2-\phi(p-1)$ for some prime number $p$ has a positive lower asymptotic density.