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Journal Articles Mathematics of Computation Year : 2014

On the number of prime factors of an odd perfect number

Pascal Ochem
Michael Rao

Abstract

Let ω(n) and Ω(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 = pᶱm² where p ≡ e ≡ 1 (mod 4), p is prime, and p ∤ m. This implies that Ω(N) ≥ 2ω(N) − 1. We. We prove that Ω(N) ≥ (18ω(N) −31) / 7andΩ(N) ≥ 2ω(N) + 51.

Dates and versions

lirmm-01263897 , version 1 (28-01-2016)

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Pascal Ochem, Michael Rao. On the number of prime factors of an odd perfect number. Mathematics of Computation, 2014, 83 (289), pp.2435-2439. ⟨10.1090/S0025-5718-2013-02776-7⟩. ⟨lirmm-01263897⟩
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