On a theorem of Erdős and Loxton

2025/04/04 by Lebowitz-Lockard, Noah
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.03446

Abstract

Let a(n) be the number of partitions of n of the form a₁ + a₂ + ⋯ + a_k where a_{i + 1} is a proper divisor of a_i for all i < k. Erd\H os and Loxton showed that the sum of a(n) over all n ≤ x is asymptotic to a constant multiple of x^ρ where s = ρ≈ 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).

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