On 2-Near Perfect Numbers

TL;DR

This paper characterizes 2-near perfect numbers, especially those of the form 2^k p^i, and explores related divisor sum properties, expanding understanding of near perfect number classifications.

Contribution

It provides a complete description of 2-near perfect numbers of specific forms and proves new results under divisor product restrictions.

Findings

Characterization of 2-near perfect numbers of the form 2^k p^i

Complete classification for these forms

Results under the condition d_1 d_2 = n

Abstract

Let $\sigma(n)$ be the sum of the positive divisors of $n$. A number $n$ is said to be 2-near perfect if $\sigma(n) = 2n +d_1 +d_2 $, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We give a complete description of those $n$ which are 2-near perfect and of the form $n=2^k p^i$ where $p$ is prime and $i \in \{1,2\}$. We also prove related results under the additional restriction where $d_1d_2=n$.