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Explicit bounds for odd perfect and quasiperfect numbers

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.NA — Numerical Analysis

Contributed by Frank Vega ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Let NN be an odd perfect number or a quasiperfect number, and let B(N)=∏p∣N(1+1/p)B(N) = \prod_{p \mid N} (1+1/p). We prove that 2<N/φ(N)<2.173722 < N/\varphi(N) < 2.17372 and that 105∤N105 \nmid N, and we obtain explicit two-sided bounds for B(N)B(N) according to the value of gcd⁡(15,N)\gcd(15,N). For an odd perfect number NN, they give B(N)>1.90281B(N) > 1.90281, 1.705841.70584, 1.832331.83233 and 1.678561.67856 when gcd⁡(15,N)=1\gcd(15,N) = 1, 33, 55 and 1515, respectively, and B(N)<1.84348B(N) < 1.84348, 1.935491.93549 and 1.841251.84125 when gcd⁡(15,N)=3\gcd(15,N) = 3, 55 and 1515. These follow from general bounds for B(n)B(n) in terms of n/φ(n)n/\varphi(n), of the abundancy index I(n)=σ(n)/nI(n) = \sigma(n)/n, and of ∑p∣n1/p\sum_{p \mid n} 1/p. We also bound the abundancy indices of Nrad⁡(N)N \operatorname{rad}(N) and N/rad⁡(N)N/\operatorname{rad}(N) and the functions σ±\sigma^{\pm}, φ+\varphi^{+} and ψ+\psi^{+} of S\'andor and Atanassov at NN, give an explicit lower bound for the number of distinct prime factors of NN in terms of its least prime factor, and show that the sum of the reciprocals of the prime factors of a quasiperfect number NN is less than log⁡I(N)\log I(N). Finally, for the shifted abundancy indices I+(n)=1+∑d∣n, d>11/(d+1)I^{+}(n) = 1 + \sum_{d \mid n, \, d > 1} 1/(d+1) and I−(n)=1+∑d∣n, d>11/(d−1)I^{-}(n) = 1 + \sum_{d \mid n, \, d > 1} 1/(d-1), we determine the optimal constants in the comparisons of I+I^{+}, II and I−I^{-} on the odd integers and on the odd integers with at most one odd exponent, and we deduce 1.8471<I+(N)<2<I−(N)<2.27221.8471 < I^{+}(N) < 2 < I^{-}(N) < 2.2722 and I−(N)−I+(N)<0.42493I^{-}(N) - I^{+}(N) < 0.42493 for every odd perfect number NN.

Provenance statement

The author used Claude (Anthropic) to write the verification script experiments.py of Section 20, to generate parts of the text of this manuscript, and in working out and checking several of the arguments. The experiments were run by the author. After using this tool, the author reviewed and edited the code and the content as needed and takes full responsibility for the content of the publication.
AI provenance filesexperiments.py · 45 KB

Tools used

Anthropic
Claude FableVersion 5.2

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Version history

  1. v1 · 32 pagesSubmitted by Frank VegaInitial depositCurrentNov 11, 2026