On a problem of Oppenheim concerning ''Factorisatio Numerorum''
type: Item
instance of: scholarly article
title: On a problem of Oppenheim concerning ''Factorisatio Numerorum''
zbMATH Open document ID: 0513.10043
DOI: 10.1016/0022-314X(83)90002-1
author: E. Rodney Canfield
author: Carl Pomerance
author: P. Erdős
published in: Journal of Number Theory
publication date: 1983-01-01T00:00:00Z
review text: Denote by \(f(n)\) the number of factorizations of a positive integer \(n\) into factors exceeding 1, the order of the factors being immaterial. In this interesting paper, the authors establish a good estimate for the maximal order of \(f(n)\), thus correcting a result of \textit{A.Oppenheim} [J. Lond. Math. Soc. 1, 205--211 (1926; JFM 52.0163.02); ibid. 2, 123--130 (1927; JFM 53.0157.02)]; their estimate is of the form \[ n\exp(-\log n(\log n)^{-1}\log_3n(1+E(n))) \] where \(E(n)=o(1)\) as \(n\to\infty\) and is given rather more explicitly in the paper, and where \(\log_k n\) denotes the \(k\)-fold iterated logarithm. A new lower bound for \(\Psi(x,x^{1/u})\), the number of positive integers \(n\leq x\) with no prime factor exceeding \(x^{1/u}\), is also derived (and applied), namely \[ \Psi(x,x^{1/u})\geq x\exp\left(-u\left(\log u+(\log_2u-1)\left(1+\frac{1}{\log u}\right)+ \mathcal O(\log_2^2u\log^{-2}u)\right)\right) \] for \(x\geq 1\), \(u\geq 3\). The paper concludes with an investigation of the largest prime divisors of highly factorable numbers n, i.e. those \(n\) for which \(f(m)< f(n)\) whenever \(m< n\) (in which case \(f(n)\) has maximal order). The 118 highly factorable numbers up to \(10^9\) are listed, and the algorithm used to obtain them described. Some additional questions are raised.
reviewed by: Eira J. Scourfield
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 3809670
zbMATH Keywords: number of factorizations of positive integer
zbMATH Keywords: maximal order
zbMATH Keywords: lower bound for Psi-function
zbMATH Keywords: largest prime divisors of highly factorable numbers
Wikidata QID: Q105517231
MaRDI profile type: Publication
cites work: Q5805143
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cites work: On Integers All of Whose Prime Factors are Small
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cites work: The Difference between Consecutive Prime Numbers
full work available at URL: 0022-314x(83)90002-1
OpenAlex ID: W2123355614
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description: scientific article; zbMATH DE number 3809670