Hyperharmonic integers exist

Doğa Can Sertbaş

doi:10.5802/crmath.137

Théorie des nombres

Hyperharmonic integers exist
[Des entiers hyperharmoniques existent]

Doğa Can Sertbaş ¹

¹ Department of Mathematics, Faculty of Sciences, Sivas Cumhuriyet University, 58140, Sivas, TURKEY.

Comptes Rendus. Mathématique, Volume 358 (2020) no. 11-12, pp. 1179-1185.

Résumé

We show that there exist infinitely many hyperharmonic integers, and this refutes a conjecture of Mező. In particular, for $r = 64 \cdot (2^{α} - 1) + 32$ , the hyperharmonic number $h_{33}^{(r)}$ is integer for 153 different values of $α (mod 748 440)$ , where the smallest $r$ is equal to $64 \cdot (2^{2659} - 1) + 32$ .

Reçu le : 2020-02-12
Révisé le : 2020-07-20
Accepté le : 2020-10-23
Publié le : 2021-01-25

DOI : 10.5802/crmath.137

Classification : 11B83, 05A10, 11B75

Affiliations des auteurs :

Doğa Can Sertbaş ¹

¹ Department of Mathematics, Faculty of Sciences, Sivas Cumhuriyet University, 58140, Sivas, TURKEY.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Do\u{g}a Can Sertba\c{s}},
     title = {Hyperharmonic integers exist},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1179--1185},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {11-12},
     year = {2020},
     doi = {10.5802/crmath.137},
     language = {en},
}

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Doğa Can Sertbaş. Hyperharmonic integers exist. Comptes Rendus. Mathématique, Volume 358 (2020) no. 11-12, pp. 1179-1185. doi : 10.5802/crmath.137. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.137/

Bibliographie
Cité par

[1] Rachid Ait Amrane; Hacène Belbachir Non-integerness of class of hyperharmonic numbers, Ann. Math. Inform., Volume 37 (2010), pp. 7-11 | MR | Zbl

[2] Rachid Ait Amrane; Hacène Belbachir Are the hyperharmonics integral? A partial answer via the small intervals containing primes, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 3-4, pp. 115-117 | DOI | MR | Zbl

[3] Emre Alkan; Haydar Göral; Doğa Can Sertbaş Hyperharmonic numbers can be rarely integers, Integers, Volume 18 (2018), A43 | Zbl

[4] John Horton Conway; Richard K. Guy The Book of Numbers, Springer, 1996 | Zbl

[5] Haydar Göral; Doğa Can Sertbaş Almost all Hyperharmonic Numbers are not Integers, J. Number Theory, Volume 171 (2017), pp. 495-526 | DOI | MR | Zbl

[6] Haydar Göral; Doğa Can Sertbaş Divisibility Properties of Hyperharmonic Numbers, Acta Math. Hung., Volume 154 (2018) no. 1, pp. 147-186 | DOI | MR | Zbl

[7] Jozsef Kürschák Über die harmonische Reihe, Mat. Fiz. Lapok, Volume 27 (1918), pp. 299-300 | Zbl

[8] István Mező About the non-integer property of hyperharmonic numbers, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Volume 50 (2007), pp. 13-20 | MR | Zbl

[9] Doğa Can Sertbaş GitHub Repository for Hyperharmonic Integers, 2020 (https://github.com/dsertbas/hyperharmonic-integers)

[10] The Sage Developers SageMath, the Sage Mathematics Software System (Version 8.3), 2018 (https://www.sagemath.org)

[11] Leopold Theisinger Bemerkung über die harmonische reihe, Monatsh. Math. Phys., Volume 26 (1915), pp. 132-134 | DOI | Zbl

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