On consecutive abundant numbers

TL;DR

This paper investigates the behavior of the maximum length of consecutive abundant numbers up to x, showing that the ratio of this length to log log log x converges to a limit between 3 and 4 as x approaches infinity.

Contribution

It proves that the ratio of the maximum consecutive abundant numbers to log log log x converges to an explicit limit, resolving a long-standing problem posed by Erdős.

Findings

The ratio E(x)/log log log x tends to a limit as x approaches infinity.

The limit value of this ratio is explicitly characterized and lies between 3 and 4.

The result confirms the conjectured growth rate of consecutive abundant numbers.

Abstract

A positive integer $n$ is called an abundant number if $\sigma (n)\ge 2n$, where $\sigma (n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H os proved that there are two positive constants $c_1$ and $c_2$ such that $c_1\log\log\log x\le E(x)\le c_2\log\log\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\log \log\log x$ tends to a limit as $x\to +\infty$, and the limit value has an explicit form which is between $3$ and $4$.