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Abstract

Suppose that $N$ is an odd perfect number and $q^\alpha$ is a prime power with $q^{\alpha}\,\|\, N $. Define the index $m= \sigma(N/q^\alpha)/q^\alpha$. We prove that $m$ cannot take the form $p^{2u}$, where $u$ is a positive integer and $2u+1$ is composite. We also prove that, if $q$ is the Euler prime, then $m$ cannot take any of the 30 forms $q_1$, $q_1^2$, $q_1^3$, $q_1^4$, $q_1^5$, $q_1^6$, $q_1^7$, $q_1^8$, $q_1q_2$, $q_1^2q_2$, $q_1^3q_2$, $q_1^4 q_2$, $q_1^5q_2$, $q_1^2q_2^2$, $q_1^3q_2^2$, $q_1^4q_2^2$, $q_1q_2q_3$, $q_1^2q_2q_3$, $q_1^3q_2q_3$, $q_1^4q_2q_3$, $q_1^2q_2^2q_3$, $q_1^2q_2^2q_3^2$, $q_1q_2q_3q_4$, $q_1^2q_2q_3q_4$, $q_1^3q_2q_3q_4$, $q_1^2q_2^2q_3q_4$, $q_1q_2q_3q_4q_5$, $q_1^2q_2q_3q_4q_5$, $q_1q_2q_3q_4q_5q_6$, $q_1q_2q_3q_4q_5q_6q_7$, where $q_1$, $q_2$, $q_3$, $q_4$, $q_5$, $q_6$, $q_7$ are distinct odd primes. A similar result is proved if $q$ is not the Euler prime. These extend recent results of Broughan, Delbourgo, and Zhou. We also pose a related problem.