Integers with a given number of divisors

TextFor any positive integer n  , let n=q1⋯qsn=q1⋯qs be the prime factorization of n   with q1≥⋯≥qs>1q1≥⋯≥qs>1. A positive integer n is said to be ordinary if the smallest positive integer with exactly n   divisors is p1q1−1⋯psqs−1, where pkpk denotes the k  th prime. Let ⌊x⌋⌊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⌊s⌋≥9(logs)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.VideoFor a video summary of this paper, please visit http://youtu.be/UeIMWjRFUnA.