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On inequalities involving counts of the prime factors of an odd perfect number

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arxiv 2303.11974 v1 pith:UJDRPISB submitted 2023-03-21 math.NT

Graeme Clayton , Cody S. Hansen This is my paper · ORCID
classification math.NT
keywords omegafracnumberfactorsprimeperfectcountingcounts
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abstract

Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number (counting multiplicity) of prime factors of $N$. We prove that $\frac{99}{37}\omega(N) - \frac{187}{37} \leq \Omega(N)$ and that if $3\nmid N$, then $\frac{51}{19}\omega(N)-\frac{46}{19} \leq \Omega(N)$.

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  1. Certified Minimal-Prime Branch Closures for Odd Perfect Numbers

    math.GM 2026-07 conditional novelty 6.5 of 10

    Relative to frozen certificate release C-small-2026-07, no odd perfect number has minimal prime divisor in {5,7,11,13,17}.

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