Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.4

The Abundancy Index of Divisors of Odd Perfect Numbers


Jose Arnaldo B. Dris
De La Salle University
Manila 1004
The Philippines

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}$.

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Received January 12 2011; revised versions received June 5 2011; October 26 2011; December 11 2011; March 23 2012. Published in Journal of Integer Sequences, April 9 2012.

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