## On the total number of prime factors of an odd perfect number

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- by D. E. Iannucci and R. M. Sorli;
- Math. Comp.
**72**(2003), 2077-2084 - DOI: https://doi.org/10.1090/S0025-5718-03-01522-9
- Published electronically: May 8, 2003
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## Abstract:

We say $n\in {\mathbb N}$ is*perfect*if $\sigma (n)=2n$, where $\sigma (n)$ denotes the sum of the positive divisors of $n$. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form $n=p^{\alpha }\prod _{j=1}^{k}q_j^{2\beta _j}$, where $p$, $q_1$, …, $q_k$ are distinct primes and $p\equiv \alpha \equiv 1\pmod 4$. We prove that if $\beta _j\equiv 1\pmod 3$ or $\beta _j\equiv 2\pmod 5$ for all $j$, $1\le j\le k$, then $3\nmid n$. We also prove as our main result that $\Omega (n)\ge 37$, where $\Omega (n)=\alpha +2\sum _{j=1}^{k}\beta _j$. This improves a result of Sayers $( \Omega (n)\ge 29 )$ given in 1986.

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## Bibliographic Information

**D. E. Iannucci**- Affiliation: University of the Virgin Islands, St. Thomas, Virgin Islands 00802
- Email: diannuc@uvi.edu
**R. M. Sorli**- Affiliation: Department of Mathematical Sciences, University of Technology, Sydney, Broadway, 2007, Australia
- Email: rons@maths.uts.edu.au
- Received by editor(s): November 7, 2001
- Published electronically: May 8, 2003
- Additional Notes: The authors are grateful for the advice and assistance given by Graeme Cohen
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp.
**72**(2003), 2077-2084 - MSC (2000): Primary 11A25, 11Y70
- DOI: https://doi.org/10.1090/S0025-5718-03-01522-9
- MathSciNet review: 1986824