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

TL;DR

We study odd perfect numbers and the broader class of odd primitive non-deficient numbers through the lens of $T(n)=\sum_{p|n} 1/p$ and $H(n)=\prod_{p|n} \frac{p}{p-1}$, relating them to the abundancy index $h(n)=\sigma(n)/n$ and the arithmetic derivative context. The authors derive new, explicit bounds in terms of the smallest prime factor $p_1$ and the surplus $S(n)=H(n)-2$, notably $T(n) \ge \log 2 - \frac{25}{64 p_1} + \frac{S}{2}-\frac{S^2}{4}$ and, under certain hypotheses, $T(n) < \log 2 - \frac{11}{50 p_1^2}$, along with $H(n) \le 2 + \frac{9}{4 p_1^2}$ for odd perfect numbers. The proofs combine analytic inequalities, precise exponential/log comparisons, and explicit prime-sum bounds (e.g., Rosser–Schoenfeld), and they extend to odd primitive non-deficient numbers. These results sharpen structural constraints on the prime-factor configurations of odd perfect numbers and contribute to the broader abundancy-oriented approach to the odd perfect number problem.

Abstract

Write $T(n)$ as the sum of the reciprocals of the primes which divide $n$. Write $H(n) = \prod_{p|n}p/(p-1)$ 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.

Theorems & Definitions (42)

• Definition 1
• Proposition 1
• Proposition 2
• Lemma 3
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• proof
• Lemma 8