On odd deficient-perfect numbers with four distinct prime divisors

TL;DR

This paper characterizes the unique odd deficient-perfect number with four distinct prime factors as 3^2 * 7^2 * 11^2 * 13^2, expanding understanding of special divisor sum properties.

Contribution

It proves that the only odd deficient-perfect number with four distinct prime divisors is a specific composite number, providing a complete classification.

Findings

The unique odd deficient-perfect number with four prime divisors is 3^2 * 7^2 * 11^2 * 13^2.

No other odd deficient-perfect numbers with four prime divisors exist.

The result narrows the search for odd deficient-perfect numbers to this specific case.

Abstract

For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma(n)=2n-d$. In this paper, we show that the only odd deficient-perfect number with four distinct prime divisors is $3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}$.