On a problem of Oppenheim concerning ''Factorisatio Numerorum''
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.
number of factorizations of positive integer
largest prime divisors of highly factorable numbers
scientific article; zbMATH DE number 3809670