Théorie des nombres
Hyperharmonic integers exist
[Des entiers hyperharmoniques existent]
Comptes Rendus. Mathématique, Tome 358 (2020) no. 11-12, pp. 1179-1185.

We show that there exist infinitely many hyperharmonic integers, and this refutes a conjecture of Mező. In particular, for r=64·(2 α -1)+32, the hyperharmonic number h 33 (r) is integer for 153 different values of α(mod748440), where the smallest r is equal to 64·(2 2659 -1)+32.

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DOI : 10.5802/crmath.137
Classification : 11B83, 05A10, 11B75
Sertbaş, Doğa Can 1

1 Department of Mathematics, Faculty of Sciences, Sivas Cumhuriyet University, 58140, Sivas, TURKEY.
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Sertbaş, Doğa Can. Hyperharmonic integers exist. Comptes Rendus. Mathématique, Tome 358 (2020) no. 11-12, pp. 1179-1185. doi : 10.5802/crmath.137. http://www.numdam.org/articles/10.5802/crmath.137/

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

[2] Ait Amrane, Rachid; Belbachir, Hacène 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] Alkan, Emre; Göral, Haydar; Sertbaş, Doğa Can Hyperharmonic numbers can be rarely integers, Integers, Volume 18 (2018), A43 | Zbl

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

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

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

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

[8] Mező, István 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] Sertbaş, Doğa Can 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] Theisinger, Leopold Bemerkung über die harmonische reihe, Monatsh. Math. Phys., Volume 26 (1915), pp. 132-134 | DOI | Zbl

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