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On the number of prime factors of an odd perfect number

Authors: Pascal Ochem and Michaël Rao
Journal: Math. Comp. 83 (2014), 2435-2439
MSC (2010): Primary 11A25, 11A51
Published electronically: November 20, 2013
MathSciNet review: 3223339
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Abstract: Let $ \Omega (n)$ and $ \omega (n)$ denote, respectively, the total number of prime factors and the number of distinct prime factors of the integer $ n$. Euler proved that an odd perfect number $ N$ is of the form $ N=p^em^2$ where $ p\equiv e\equiv 1\pmod 4$, $ p$ is prime, and $ p\nmid m$. This implies that $ \Omega (N)\ge 2\omega (N)-1$. We prove that $ \Omega (N)\ge (18\omega (N)-31)/7$ and $ \Omega (N)\ge 2\omega (N)+51$.

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Pascal Ochem
Affiliation: CNRS, LIRMM, Université Montpellier 2, 161 rue Ada, 34095 Montpellier Cedex 5, France

Michaël Rao
Affiliation: CNRS, LIP, ENS Lyon, 15 parvis R. Descartes BP 7000, 69342 Lyon Cedex 07, France

Received by editor(s): September 15, 2012
Received by editor(s) in revised form: December 18, 2012
Published electronically: November 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.