Are the hyperharmonics integral? A partial answer via the small intervals containing primes

Rachid Aït Amrane; Hacène Belbachir

doi:10.1016/j.crma.2010.12.015

Combinatorics/Algebra

Are the hyperharmonics integral? A partial answer via the small intervals containing primes

Presented by: the Editorial Board

Rachid Aït Amrane ¹; Hacène Belbachir ²

¹ ESI/École nationale supérieure d'informatique, BP 68M, Oued Smar, 16309, El Harrach, Alger, Algeria
² USTHB, faculté de mathématiques, BP 32, El Alia, 16111 Bab Ezzouar, Alger, Algeria

Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 115-117.

Abstract (Original language)
Résumé

In a recent work, the authors have used Bertrand's postulate to give a partial answer to the conjecture of Mező which says that the hyperharmonic numbers – iterations of partial sums of harmonic numbers – are not integers. In this Note, using small intervals containing prime numbers, we prove that a great class of hyperharmonic numbers are not integers.

Dans un travail antérieur, les auteurs ont utilisé le postulat de Bertrand pour répondre, partiellement, à la conjecture de Mező selon laquelle les nombres hyperharmoniques – itérations de sommes partielles de nombres harmoniques – ne sont pas des entiers. Dans cette Note, nous montrons qu'une grande classe de nombres hyperharmoniques ne sont pas des entiers en utilisant les petits intervalles contenant des nombres premiers.

Received: 2010-10-26
Accepted: 2010-12-23
Published online: 2011-01-05

DOI: 10.1016/j.crma.2010.12.015

Author's affiliations:

Rachid Aït Amrane ¹; Hacène Belbachir ²

¹ ESI/École nationale supérieure d'informatique, BP 68M, Oued Smar, 16309, El Harrach, Alger, Algeria
² USTHB, faculté de mathématiques, BP 32, El Alia, 16111 Bab Ezzouar, Alger, Algeria

@article{CRMATH_2011__349_3-4_115_0,
     author = {Rachid A{\"\i}t Amrane and Hac\`ene Belbachir},
     title = {Are the hyperharmonics integral? {A} partial answer via the small intervals containing primes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {115--117},
     publisher = {Elsevier},
     volume = {349},
     number = {3-4},
     year = {2011},
     doi = {10.1016/j.crma.2010.12.015},
     language = {en},
}

TY  - JOUR
AU  - Rachid Aït Amrane
AU  - Hacène Belbachir
TI  - Are the hyperharmonics integral? A partial answer via the small intervals containing primes
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 115
EP  - 117
VL  - 349
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2010.12.015
LA  - en
ID  - CRMATH_2011__349_3-4_115_0
ER  -

%0 Journal Article
%A Rachid Aït Amrane
%A Hacène Belbachir
%T Are the hyperharmonics integral? A partial answer via the small intervals containing primes
%J Comptes Rendus. Mathématique
%D 2011
%P 115-117
%V 349
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2010.12.015
%G en
%F CRMATH_2011__349_3-4_115_0

Rachid Aït Amrane; Hacène Belbachir. Are the hyperharmonics integral? A partial answer via the small intervals containing primes. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 115-117. doi : 10.1016/j.crma.2010.12.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.12.015/

References
Cited by

[1] R. Aït Amrane; H. Belbachir Non-integerness of class of hyperharmonic numbers, Ann. Mathematicae et Informaticae, Volume 37 (2010), pp. 7-11

[2] D. Bazzanella; A. Languasco; A. Zaccagnini Prime numbers in logarithmic intervals, 17 Sept. 2008 | arXiv

[3] J.H. Conway; R.K. Guy The Book of Numbers, Springer-Verlag, New York, 1996

[4] G. Giordano The generalization and proof of Bertrand's Postulate, Internat. J. Math. & Math. Sci., Volume 10 (1987) no. 4, pp. 821-824

[5] C. Jia Almost all short intervals containing prime numbers, Acta Arith., Volume LXXVLI (1996)

[6] N. Jitsuro On the interval containing at least one prime number, Proc. Japan Acad., Volume 28 (1952), pp. 177-181

[7] I. Mező About the non-integer property of hyperharmonic numbers, Ann. Univ. Sci. Budapest, Sect. Math., Volume 50 (2007), pp. 13-20

[8] O. Ramaré; Y. Saouter Short effective intervals containing primes, J. Number Theory, Volume 98 (2003), pp. 10-33

[9] L. Schoenfeld Sharper bounds for the Chebyshev functions $θ (x)$ and $ψ (x)$ . II, Math. Comp., Volume 30 (1976) no. 134, pp. 337-360

[10] L. Taeisinger Bemerkung über die harmonische Reihe, Monatsch. Math. Phys., Volume 26 (1915), pp. 132-134

Cited by Sources:

Comments - Policy