Quasiperfect numbers with the same exponent

TL;DR

This paper investigates the divisibility properties of quasiperfect numbers, establishing conditions on their exponents and prime factors, and providing bounds for specific forms of these numbers.

Contribution

It introduces new divisibility criteria for quasiperfect numbers and bounds on their prime factors, advancing understanding of their structure.

Findings

If $N=(p_1 p_2 imes ext{...})^{2a}$ is quasiperfect, then $2a+1$ is divisible by 3.

A quasiperfect number $N=m^2$ has at least one prime factor smaller than $ ext{exp}(716.7944)$.

Lower bounds are established for quasiperfect numbers of the form $N=m^2$ with $m$ squarefree.

Abstract

We study some divisibility properties of quasiperfect numbers. We show that if $N=(p_1 p_2 \cdots p_t)^{2a}=m^2$ is quasiperfect, then $2a+1$ is divisible by $3$ and $N$ has at least one prime factor smaller than $\exp 716.7944$. Moreover, we find some lower bounds concerning quasiperfect numbers of the form $N=m^2$ with $m$ squarefree.