On the smallest number with a given number of divisors

Abstract

For a natural number \(n\), let \(A(n)\) denote the smallest natural number that has exactly \(n\) divisors. Let \(p_{1}p_{2}\ldots p_{k}\) be the prime decomposition of \(n\) where the primes are given in decreasing order and the factors are not necessarily distinct. If \(q_{1},\ldots ,q_{k}\) denote the first \(k\) primes and \(A(n)=q_{1}^{p_{1}-1}\ldots q_{k}^{p_{k}-1}\), we say that \(n\) is ordinary. If \(n\) is not ordinary, we say it is extraordinary. In Brown (2006), it was shown that almost all numbers are ordinary and if \(E_{x}\) denotes the set of extraordinary numbers less than or equal to a positive real number \(x\), \(|E_{x}|=o\left( \frac{x}{2^{(\log (\log x))^{\delta }}}\right) \) for any \(\displaystyle 0<\delta <1/2\). In what follows, we will prove that \(\displaystyle |E_x|=\frac{x}{\log x}e^{\Psi (x)}\) where \(\displaystyle \Psi (x)\sim \frac{1}{2\log 2}\frac{\log _{(3)}^2x}{\log _{(4)}x}\) and \(\displaystyle \log _{(r)}x\) denotes the \(r\)-times iterated natural logarithm. We make use of the Prime Number Theorem as well as estimates of the size of sets where the number of total prime factors or distinct prime factors is fixed.