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Let $N$ be an odd perfect number. Then, Euler proved that there exist someintegers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmidn$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that theratio $\frac{\sigma(n^{2})}{q^{\alpha}}$ is neither a square nor a square timesa single prime unless $\alpha = 1$. It is a direct consequence of a certainproperty of the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$, where$l$ denotes one or a prime, whose proof is based on the prime idealfactorization in the quadratic orders $\mathbb{Z}[\sqrt{1-q}]$ and theprimitive solutions of generalized Fermat equations $x^{\beta}+y^{\beta} =2z^{2}$. We give also a slight generalization to odd multiply perfect numbers.

A note on the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$ and application to odd perfect numbers