  We parameterize solutions to the equality
$\Phi_3(x)=\Phi_3(a_1)\Phi_3(a_2)\cdots\Phi_3(a_n)$ when each $\Phi_3(a_i)$ is
prime. Our focus is on the special cases when $n=2,3,4$, as this analysis
simplifies and extends bounds on the total number of prime factors of an odd
perfect number.
