The distribution of numbers with many ordered factorizations
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 583-606.

Let $g\left(n\right)$ be the number of ordered factorizations of $n$ into numbers larger than $1$. We find precise bounds on the positive moments of $g$. We use these results to estimate the number of $n\le x$ satisfying $g\left(n\right)\ge {x}^{\alpha }$ for all positive $\alpha$. In addition, let $G\left(n\right)$ and ${g}_{𝒫}\left(n\right)$ be the number of ordered factorizations of $n$ into distinct numbers larger than $1$ and primes, respectively. We also bound the positive moments of $G$ and ${g}_{𝒫}$ from below.

Soit $g\left(n\right)$ le nombre de factorisations de $n$ en produit ordonné de facteurs plus grands que $1$. On trouve des bornes précises pour les moments positifs de $g$. On utilise ces résultats pour estimer le nombre de $n\le x$ tels que $g\left(n\right)\ge {x}^{\alpha }$ pour tous les $\alpha$ positifs. En outre, soient $G\left(n\right)$ et ${g}_{𝒫}\left(n\right)$ les nombres de factorisations de $n$ en produit ordonné de facteurs distincts plus grands que $1$ et en produit ordonné de facteurs premiers respectivement. On donne des bornes inférieures pour les moments positifs de $G$ et ${g}_{𝒫}$.

Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1170
Classification: 11A25, 11A51, 11N37
Keywords: Ordered factorizations
Lebowitz-Lockard, Noah 1

1 8330 Millman St. Philadelphia, PA, 19118, United States
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Lebowitz-Lockard, Noah. The distribution of numbers with many ordered factorizations. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 583-606. doi : 10.5802/jtnb.1170. http://archive.numdam.org/articles/10.5802/jtnb.1170/

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