A new approach to odd perfect numbers via GCDs

Abstract

Let $q^k{n}^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G=\gcd (\sigma (q^k),\sigma (n^2))$$ $$H=\gcd (n^2,\sigma (n^2))$$ and $$I=\gcd (n,\sigma (n^2)).$$ We prove that $G\times H=I^2$. (Note that it is trivial to show that $G\mid I$ and $I\mid H$ both hold.) We then compute expressions for $G,H,$ and $I$ in terms of $\sigma (q^k)/2,n,$ and $\gcd (\sigma (q^k)/2,n)$. Afterwards, we prove that if $G=H=I$, then $\sigma (q^k)/2$ is not squarefree. Other natural and related results are derived further. Lastly, we conjecture that the set $$\mathcal{A}=\{m:\gcd (m,\sigma (m^2))=\gcd (m^2,\sigma (m^2))\}$$ has asymptotic density zero.

9 pages