On near-perfect numbers
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 of the well-known pseudoperfect numbers first defined by Sierpiński, and are numbers that equal the sum of all but at most $k$ of their proper divisors. We establish the asymptotic order of $k$-near-perfect numbers for all integers $k\ge 4$, as well as some properties of related quantities.