Must a primitive non-deficient number have a component not much larger than its radical?

Technical Summary: "Must a Primitive Non-Deficient Number Always Have a Small Component?"

Problem Statement
The paper investigates the structural properties of primitive non-deficient numbers. A positive integer $n$ is non-deficient if $\sigma(n) \ge 2n$, where $\sigma(n)$ is the sum of its divisors. A number is primitive non-deficient if it is non-deficient but all of its proper divisors are deficient. Let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ be the prime factorization of such a number, with $p_1 < p_2 < \dots < p_k$, and let $R = \text{rad}(n) = p_1 p_2 \cdots p_k$ be its radical.

The central problem is to determine bounds on the "components" of $n$, specifically the terms $p_i^{a_i+1}$. The author posits that for a primitive non-deficient number to exist, at least one of its prime power components must be relatively small compared to the radical $R$. This is motivated by the intuition that if all exponents $a_i$ were large, the abundancy index $h(n) = \sigma(n)/n$ would be extremely close to the upper bound $H(n) = \prod_{p|n} \frac{p}{p-1}$. Consequently, reducing any exponent would likely leave the number non-deficient, contradicting the primitivity of $n$.

Methodology
The paper employs analytic number theory techniques, focusing on the properties of the abundancy index $h(n)$ and the function $H(n)$. The methodology proceeds through the following logical steps:

1. Foundational Lemmas: The author establishes basic properties of non-deficient numbers, including bounds on the number of distinct prime factors $k$ relative to the smallest prime factor $p_1$ (Lemma 5, adapted from Grun and Servais).
2. Inequality Derivation: A key technical tool is Lemma 9, which provides an upper bound for $\min(p_i^{a_i+1})$ given a lower bound on $H(n)$. This lemma adapts arguments from Heath-Brown, utilizing the relationship between $h(n)$ and $H(n)$:
   $$h(n) = H(n) \prod_{i=1}^k \left(1 - \frac{1}{p_i^{a_i+1}}\right)$$
   By assuming $n$ is primitive, the author notes that $h(n/p_i) < 2$ for all $i$. Combining this with Lemma 8 (which bounds the ratio $h(n)/h(n/p)$), the author derives inequalities relating the minimum component size to the "gap" between $H(n)$ and 2.
3. Case Analysis: The proof distinguishes between even and odd primitive non-deficient numbers. For even numbers, Lemma 3 and Proposition 6 show that the component corresponding to the prime 2 satisfies the conjectured bound directly. For odd numbers, the author relies on Lemma 4 to handle small values of $k$ (3 and 4) via direct computation, and Lemma 9 for larger $k$.

Key Contributions and Results

• Theorem 2 (Main Result): The paper proves that for any primitive non-deficient number $n = p_1^{a_1} \cdots p_k^{a_k}$ with radical $R$, there exists an index $i$ such that:
  $$p_i^{a_i+1} < 2kR$$
  This establishes that at least one component is bounded linearly by the product of the number of distinct prime factors and the radical.

• Conjecture 1: The author proposes a stronger, unproven conjecture: that the factor $k$ in Theorem 2 is unnecessary, and that there always exists an $i$ such that:
  $$p_i^{a_i+1} < 2R$$
  The author notes that the bound $R$ (without the factor 2) is false, citing a specific counterexample ($n = 4164647056479$), but suggests the counterexample may be unique.

• Proposition 16: For odd primitive non-deficient numbers, the paper proves a dichotomy: either the largest prime factor $p_k$ is small relative to the radical ($p_k < \sqrt{R}$), or there exists a component satisfying a bound involving $p_k$ and $k$.

• Density Conjectures: The paper introduces sets $B$ (numbers satisfying $p_i^{a_i+1} < 2R$ for all $i$) and $C$ (primitive non-deficient numbers). It conjectures that the intersection of these sets has natural density zero within $B$ and within $C$, suggesting that while the "small component" property is necessary for primitivity, having all components small is a rare occurrence.

Significance and Context
The paper situates its results within the broader study of odd perfect numbers and non-deficient numbers.

• Graph Theory Connection: The bounds on $p_i^{a_i+1}$ are interpreted as bounds on the out-degree of vertices in a weighted directed graph representation of the components of odd perfect numbers, a framework used in recent literature (e.g., Bibby, Vyncke, and Zelinsky; Yamada).
• Generalization of Existing Bounds: The results generalize previous upper bounds on the size of odd perfect numbers in terms of their radical. Specifically, Nielsen's bound $n < R^{2k-1}$ implies a bound on components; Theorem 2 provides a tighter, more specific inequality applicable to the broader class of primitive non-deficient numbers.
• Motivation: The work aims to formalize the intuition that primitivity imposes strict constraints on the size of prime power components. While the paper does not resolve the existence of odd perfect numbers, it provides refined structural constraints that any such number (or any primitive non-deficient number) must satisfy.

The author acknowledges that the techniques naturally yield inequalities involving $p_i^{a_i+1}$ rather than $p_i^{a_i}$, though these are mathematically equivalent to bounding a component by the product of the other prime factors. The paper concludes by noting that while the conjectured bound $2R$ is likely true, proving it remains an open challenge.