On a theorem of Erd\H{o}s and Loxton

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 $x^\rho$ where $s = \rho \approx 1.73$ is the unique solution to the equation $\zeta(s) = 2$ 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 $a(n)$.