Integers with a given number of divisors

Text. For any positive integer n, let n = q(1) . . . q(s) be the prime factorization of n with q(1) >= . . . >= q(s) > 1. A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p(1)(q1-1) . . . p(s)(qs) (-) (1), where P-k denotes the kth prime. Let [x] be the largest integer not exceeding x. In 2006, Brown proved that all square-free integers are ordinary and the set of all ordinary integers has asymptotic density one. In this paper, we prove that, if q([root s]) >= 9(log s)(2), then n is ordinary. Furthermore, the set of such integers n has asymptotic density one. We also determine all ordinary integers which are not divisible by any fifth power of a prime. Video. For a video summary of this paper, please visit http://youtu.be/UeIMWjRFUnA. (C) 2014 Elsevier Inc. All rights reserved.