How often is the number of divisors of n a divisor of n?

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Abstract

Let d(n) denote the number of positive integers dividing the positive integer n. We show that as x approaches ∞, #{n ≤ x: d(n) divides n} = (xlog x)(log log x)−1 + 0(1).

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This paper is, with minor changes, part of the Ph.D. dissertation of the author, written at The University of Illinois at Urbana/Champaign under the direction of Paul T. Bateman.