On ordered factorizations into distinct parts
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- by Noah Lebowitz-Lockard and Paul Pollack
- Proc. Amer. Math. Soc. 148 (2020), 1447-1453
- DOI: https://doi.org/10.1090/proc/14817
- Published electronically: November 19, 2019
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Abstract:
Let $g(n)$ denote the number of ordered factorizations of $n$ into integers larger than $1$. In the 1930s, Kalmár and Hille investigated the average and maximal orders of $g(n)$. In this note we examine these questions for the function $G(n)$ counting ordered factorizations into distinct parts. Concerning the average of $G(n)$, we show that as $x\to \infty$, \[ \sum _{n \le x} G(n) = x \cdot L(x)^{1+o(1)}, \] where \[ L(x) = \exp \left (\log {x} \cdot \frac {\log \log \log {x}}{\log \log {x}}\right ). \] It follows immediately that $G(n) \le n \cdot L(n)^{1+o(1)}$, as $n\to \infty$. We show that equality holds here on a sequence of $n$ tending to infinity, so that $n \cdot L(n)^{1+o(1)}$ represents the maximal order of $G(n)$.References
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Bibliographic Information
- Noah Lebowitz-Lockard
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 1128605
- Email: nlebowi@gmail.com
- Paul Pollack
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 830585
- Email: pollack@uga.edu
- Received by editor(s): May 20, 2019
- Received by editor(s) in revised form: August 13, 2019
- Published electronically: November 19, 2019
- Additional Notes: During the writing of this paper, the second author was supported by NSF award DMS-1402268
- Communicated by: Amanda Folsom
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1447-1453
- MSC (2010): Primary 11N37; Secondary 11N64
- DOI: https://doi.org/10.1090/proc/14817
- MathSciNet review: 4069184