## Asymptotic bounds for factorizations into distinct parts

### Volume 201 / 2021

#### Abstract

Let $f(n)$ be the number of unordered factorizations of $n$ into parts greater than $1$ and let $F(n)$ be the number of such factorizations into distinct parts. For arbitrary $n$, we find new upper and lower bounds for $F(n)$ and show that these bounds are close together. Using a similar technique, we also bound from above the number of ordered factorizations into distinct parts greater than $1$. We also find a new upper bound for $f(n)$ which is similar to a lower bound of Balasubramanian and Srivastav. We also bound the ratio $f(n)/F(n)$ and use this result to obtain a constructive proof of the maximal order of $F(n)$ for $n \leq x$. Finally, we bound the number of numbers $\leq x$ which lie in the ranges of $F$ and $f$.