On consecutive abundant numbers

2016/03/20 by Yong-Gao Chen, Chen, Yong-Gao, Hui Lv +1
Mathematics · #11N37 #11N60 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N37 #msc:11N60

paper · pdf · doi:10.48550/arxiv.1603.06176

14pages

arxiv created 2016/03/20 · arxiv updated 2016/03/22

Abstract

A positive integer n is called an abundant number if σ(n)≥ 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 c₁ and c₂ such that c₁logloglog x≤ E(x)≤ c₂logloglog x. In this paper, we resolve this old problem by proving that, E(x)/log loglog x tends to a limit as x→ +∞, and the limit value has an explicit form which is between 3 and 4.

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