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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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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Additional Information
  • Pascal Ochem
  • Affiliation: CNRS, LIRMM, Université Montpellier 2, 161 rue Ada, 34095 Montpellier Cedex 5, France
  • Email:
  • Michaël Rao
  • Affiliation: CNRS, LIP, ENS Lyon, 15 parvis R. Descartes BP 7000, 69342 Lyon Cedex 07, France
  • MR Author ID: 714149
  • Email:
  • 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:
  • MathSciNet review: 3223339