On the Number of Factorizations of An Integer

Abstract

Let f (n) denote the number of unordered factorizations of a positive integer n into factors larger than 1. We show that the number of distinct values of f (n), less than or equal to x, is at most exp (C a��log x /a�loglog x (1+o(1))), where C = 2π a�� 2/3 and x is sufficiently large. This improves upon a previous result of the first author and F. Luca.