On the number of factorizations of an integer

TL;DR

This paper establishes an upper bound on the number of distinct values of the function counting unordered factorizations of integers, improving previous results and revealing growth limitations for large x.

Contribution

The paper provides a tighter upper bound on the count of distinct factorization values, advancing understanding of the distribution of the function f(n).

Findings

Upper bound on the number of distinct f(n) values is exponential in a square root of a logarithmic ratio.

The bound improves upon previous results by the first author and F. Luca.

The result applies to sufficiently large x, indicating growth constraints for the factorization function.

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 $\exp \left( C \sqrt{\frac{\log x}{\log \log x}} \left( 1 + o(1) \right) \right)$, where $C=2\pi\sqrt{2/3}$ and $x$ is sufficiently large. This improves upon a previous result of the first author and F. Luca.