On Near Perfect Numbers

TL;DR

This paper explores the properties and asymptotic behavior of k-near-perfect numbers, a generalization of perfect numbers, providing new insights into their distribution for all integers k ≥ 4.

Contribution

It establishes the asymptotic order of k-near-perfect numbers for all k ≥ 4 and investigates related properties, advancing understanding of these generalized perfect numbers.

Findings

Established asymptotic order for k-near-perfect numbers when k ≥ 4

Identified properties of numbers related to k-near-perfectness

Extended classical perfect number theory to a broader class

Abstract

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.