On the smallest number with a given number of divisors

Autor: Anna K. Savvopoulou, Christopher M. Wedrychowicz
Rok vydání: 2014
Předmět:
Zdroj: The Ramanujan Journal. 37:51-64
ISSN: 1572-9303
1382-4090
DOI: 10.1007/s11139-014-9572-9
Popis: 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
Databáze: OpenAIRE
Externí odkaz: