Complete characterization of $2$-near perfect numbers with exactly 2 prime factors

TL;DR

This paper fully characterizes 2-near perfect numbers with exactly two prime factors, proving the absence of odd such numbers and describing the structure of even ones under certain conditions.

Contribution

It provides a complete classification of 2-near perfect numbers with two prime factors, resolving open questions about their existence and structure.

Findings

No odd 2-near perfect numbers with two prime factors exist.

All even 2-near perfect numbers with certain parameters belong to a specific family.

The results complete the characterization of 2-near perfect numbers with two prime factors.

Abstract

Let $\sigma(n)$ be the sum of the positive divisors of $n$. A positive integer $n$ is said to be $2$-near perfect when $\sigma(n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We show that there are no odd $2$-near perfect numbers with exactly two prime factors, and that all even $2$-near perfect numbers (i.e. those of the form $2^kp^m$, where $p$ is an odd prime) belong to a specific family, provided that $m$ is at least 3. In combination with prior work, these results produce a complete characterization of $2$-near perfect numbers with exactly 2 prime factors.