  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.
