  The study of perfect numbers (numbers which equal the sum of their proper
divisors) goes back to antiquity, and is responsible for some of the oldest and
most popular conjectures in number theory. We investigate a generalization
introduced by Pollack and Shevelev: $k$-near-perfect numbers. These are
examples to the well-known pseudoperfect numbers first defined by Sierpi\'nski,
and are numbers such that the sum of all but at most $k$ of its proper divisors
equals the number. We establish their asymptotic order for all integers $k\ge
4$, as well as some properties of related quantities.
