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 = qkn2 is an odd perfect number, where q is the Euler prime, then we show that n < q is sufficient for Sorli's conjecture that k = νq(N) = 1 to hold. We also prove that qk < (2/3)n2, and that I(qk) < I(n), where I(x) is the abundancy index of x.

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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. Withdrawn because of errors, April 18 2012. Revised versions received April 25 2012; August 3 2012. Published in Journal of Integer Sequences, September 3 2012.

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