On near-perfect and deficient-perfect numbers
For a positive integer $n$, let $\sigma (n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a near-perfect number if $\sigma (n) = 2n + d$, and a deficient-perfect number if $\sigma (n) = 2n - d$. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.