Long gaps between deficient numbers
Paul Pollack · Acta Arithmetica · 2010
Abstract. Let n be a natural number. If the sum of the proper divisors of n is less than n, then n is said to be deficient. Let G(x) be the largest gap between consecutive deficient numbers belonging to the interval [1, x]. In 1935, Erdős proved that there are positive constants c1 and c2 with c1 log log log x ≤ G(x) ≤ c2 log log log x for all large x. We prove that G(x) / log log log x tends to a limit as x→∞, and we describe the limit explicitly in terms of the distribution function of σ(n)/n. We also prove that long runs of nondeficient numbers are scarce, in that the proportion of n for which n+1,..., n+A are all nondeficient decays triply exponentially in A. 1.