The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number

2024/02/22 by Joshua Zelinsky, Zelinsky, Joshua
Mathematics · #11A25 #11N64 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2402.14234

openalex publication_date 2024/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Write T(n) as the sum of the reciprocals of the primes which divide n. Write H(n) = ∏_p|np/(p-1) where the product is over the prime divisors of n. We prove new bounds for T(n) and H(n) in terms of the smallest prime factor of n, under the assumption that n is an odd perfect number. Some of the results also apply under the weaker assumption that n is odd and primitive non-deficient.

Related

• On the Sum and Product of Distinct Prime Factors of an Odd Perfect Number
• On the small prime factors of a non-deficient number
• New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number
• Bounds On Factors Of Odd Perfect Numbers
• A Proof of the Odd Perfect Number Conjecture