On a theorem of Erdős and Loxton

TL;DR

This work analyzes partitions of integers into distinct parts with each part dividing the previous one, linking them to multiplicative factorizations. It proves tight bounds on the Erdős–Loxton constant in the asymptotic $A(x)\sim c x^{\rho}$ by relating $A(x)$ to the known $G(x)$ via a bijection with factorization counts, and derives explicit upper bounds for $a(n)$. The approach uses a decomposition $A(x)=B(x)+B(x-1)$ with $G(x/2)\le B(x)\le G(x)$ and leverages asymptotics $G(x)\sim -\frac{1}{\rho\zeta'(\rho)} x^{\rho}$ where $\rho$ solves $\zeta(\rho)=2$. Additional results include recurrence relations for $a(n)$ and $b(n)$, and bounds on their maximal orders, as well as comparisons to related functions $g(n)$, $b(n)$, and $G(x)$, clarifying the growth landscape of these partition/factorization counts.

Abstract

Let $a(n)$ be the number of partitions of $n$ of the form $a_1 + a_2 + \cdots + a_k$ where $a_{i + 1}$ is a proper divisor of $a_i$ for all $i < k$. Erd{\H o}s and Loxton showed that the sum of $a(n)$ over all $n \leq x$ is asymptotic to a constant multiple of $x^ρ$ where $s = ρ\approx 1.73$ is the unique solution to the equation $ζ(s) = 2$ satisfying $s > 1$. In this note, we provide tight bounds on the value of this constant, though we do not find an exact formula for it. In addition, we write an explicit upper bound for $a(n)$.

Theorems & Definitions (12)

• Definition
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5: ELox
• Proposition 6
• proof
• proof : Proof of Theorem \ref{['main']}
• Theorem 7