On consecutive abundant numbers

A positive integer $n$ is called an abundant number if $σ(n)\ge 2n$, where $σ(n)$ 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 $4$.

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