Cao, Huizhong Essentially different factorizations of a natural number. (English) Zbl 0716.11049 Compos. Math. 77, No. 3, 343-346 (1991). Let f(n) denote the number of essentially different factorizations of a natural number n. J. F. Hughes and J. O. Shallit [Am. Math. Mon. 90, 468-471 (1983; Zbl 0523.10007)] have shown that \(f(n)\leq 2n^ c\), \(c=\sqrt{2}\), for all n. They made two conjectures, (1) f(n)\(\leq n\) for all n, and (2) f(n)\(\leq n/\log n\) for \(n\neq 144\). Xiaoxia Chen [Acta Math. Sin. 30, 268-271 (1987; Zbl 0624.10010)] has shown that (1) is true. In the present paper the author shows the following theorem: Let \(A>0\) be fixed, then \(f(n)\leq C^ n/(\log^ An)\) for every odd number \(n>1\), where C is a constant only related to A. Reviewer: Xuan Tizuo MSC: 11P82 Analytic theory of partitions 11A25 Arithmetic functions; related numbers; inversion formulas Keywords:essentially different factorizations Citations:Zbl 0523.10007; Zbl 0624.10010 PDFBibTeX XMLCite \textit{H. Cao}, Compos. Math. 77, No. 3, 343--346 (1991; Zbl 0716.11049) Full Text: Numdam EuDML References: [1] Hughes, J.F. and Shallit, J.O. , On the number of multiplicative partitions , Amer. Math. Monthly 90 (1983), 468-471. · Zbl 0523.10007 · doi:10.2307/2975729 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.