On the square-root partition function

Yong-Gao Chen; Ya-Li Li

doi:10.1016/j.crma.2015.01.013

Number theory

On the square-root partition function

Presented by: the Editorial Board

Yong-Gao Chen ¹; Ya-Li Li ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 287-290.

Abstract
Résumé

The well-known partition function $p (n)$ , which is the number of solutions of the equation $n = a_{1} + \dots + a_{k}$ with integers $1 \leq a_{1} \leq \dots \leq a_{k}$ , has a long research history. In this note, we investigate a new partition function. Let $q (n)$ be the number of solutions of the equation $n = [\sqrt{a_{1}}] + \dots + [\sqrt{a_{k}}]$ with integers $1 \leq a_{1} \leq \dots \leq a_{k}$ , where $[x]$ denotes the integral part of x. We prove that $\exp (c_{1} n^{2 / 3}) \leq q (n) \leq \exp (c_{2} n^{2 / 3})$ for two explicit positive constants $c_{1}$ and $c_{2}$ .

La fonction de partition bien connue $p (n)$ , qui compte le nombre de solutions de l'équation $n = a_{1} + \dots + a_{k}$ en entiers $1 \leq a_{1} \leq \dots \leq a_{k}$ , a une longue histoire. Nous étudions dans cette Note une nouvelle fonction de partition. Soit $q (n)$ le nombre de solutions de l'équation $n = [\sqrt{a_{1}}] + \dots + [\sqrt{a_{k}}]$ en entiers $1 \leq a_{1} \leq \dots \leq a_{k}$ , où $[x]$ désigne la partie entière de x. Nous montrons que $\exp (c_{1} n^{2 / 3} \leq q (n) \leq \exp (c_{2} n^{2 / 3})$ pour deux constantes positives explicites $c_{1}$ et $c_{2}$ .

Received: 2014-10-09
Accepted: 2015-01-27
Published online: 2015-02-13

DOI: 10.1016/j.crma.2015.01.013

Author's affiliations:

Yong-Gao Chen ¹; Ya-Li Li ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

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     author = {Yong-Gao Chen and Ya-Li Li},
     title = {On the square-root partition function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {287--290},
     publisher = {Elsevier},
     volume = {353},
     number = {4},
     year = {2015},
     doi = {10.1016/j.crma.2015.01.013},
     language = {en},
}

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Yong-Gao Chen; Ya-Li Li. On the square-root partition function. Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 287-290. doi : 10.1016/j.crma.2015.01.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.013/

References
Cited by

[1] R. Balasubramanian; F. Luca On the number of factorizations of an integer, Integers, Volume 11 (2011) (A12, 5 p)

Cited by Sources:

^☆ This work was supported by the National Natural Science Foundation of China (No. 11371195) and PAPD.

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