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The Abundancy Index of Divisors of
Odd Perfect Numbers Jose Arnaldo B. Dris
De La Salle University
Manila 1004
The Philippines
email:jabdris@yahoo.com.phjabdris@yahoo.com.ph

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Abstract:

If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $\sigma(n) \le q^k$ is necessary and sufficient for Sorli's conjecture that $k = \nu_{q}(N) = 1$ to hold. It follows that, if $k = 1$ then the Euler prime $q$ is the largest prime factor of $N$ and that $q > {10}^{500}$. We also prove that $q^k < \frac{2}{3}{n^2}$.