Bounds for Odd $k$-Perfect Numbers

TL;DR

This paper establishes bounds on the number of odd k-perfect numbers with a limited number of prime factors, contributing to the understanding of their distribution and rarity.

Contribution

It provides a new upper bound on the count of odd k-perfect numbers with a given number of prime factors, advancing the theoretical understanding of these numbers.

Findings

Number of odd k-perfect numbers with at most r prime factors is bounded by k4^{r^3}

The result applies for all integers r ≥ 1 and k ≥ 2

Advances the theoretical bounds on the distribution of odd k-perfect numbers

Abstract

Let $k\ge2$ be an integer. A natural number $n$ is called $k$-perfect if $\sigma(n)=kn.$ For any integer $r\ge1$ we prove that the number of odd $k$-perfect numbers with at most $r$ distinct prime factors is bounded by $k4^{r^3}$.