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Asymptotic bounds for factorizations into distinct parts

Volume 201 / 2021

Noah Lebowitz-Lockard Acta Arithmetica 201 (2021), 371-389 MSC: Primary 11N37; Secondary 11N56. DOI: 10.4064/aa200715-15-10 Published online: 24 November 2021


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 $f$.


  • Noah Lebowitz-Lockard8330 Millman St.
    Philadelphia, PA 19118, U.S.A.

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