On the number of prime factors of an odd perfect number
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- by Pascal Ochem and Michaël Rao PDF
- Math. Comp. 83 (2014), 2435-2439 Request permission
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 $\Omega (N)\ge 2\omega (N)+51$.References
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Additional Information
- Pascal Ochem
- Affiliation: CNRS, LIRMM, Université Montpellier 2, 161 rue Ada, 34095 Montpellier Cedex 5, France
- Email: ochem@lirmm.fr
- Michaël Rao
- Affiliation: CNRS, LIP, ENS Lyon, 15 parvis R. Descartes BP 7000, 69342 Lyon Cedex 07, France
- MR Author ID: 714149
- Email: michael.rao@ens-lyon.fr
- Received by editor(s): September 15, 2012
- Received by editor(s) in revised form: December 18, 2012
- Published electronically: November 20, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 2435-2439
- MSC (2010): Primary 11A25, 11A51
- DOI: https://doi.org/10.1090/S0025-5718-2013-02776-7
- MathSciNet review: 3223339