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

TL;DR

This paper investigates the properties of primitive non-deficient numbers, establishing bounds on prime powers relative to their product, and conjectures a tighter inequality involving prime components.

Contribution

It proves a bound on prime powers in primitive non-deficient numbers and proposes a conjecture for a sharper inequality.

Findings

Existence of an index i with p_i^{a_i+1} < 2k times the product of primes

Conjecture that p_i^{a_i+1} < product of all primes for some i

Provides bounds relating prime exponents to the number's prime product

Abstract

Let $n$ be a primitive non-deficient number where $n=p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1, p_2 \cdots p_k$ are distinct primes. We prove that there exists an $i$ such that $$p_i^{a_i+1} < 2k(p_1p_2p_3\cdots p_k).$$ We conjecture that in fact one can always find an $i$ such that ${p_i}^{a_i+1} < p_1p_2p_3\cdots p_k$.