Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-30T02:06:15.519Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE INDEX OF COMPOSITION OF THE EULER FUNCTION AND OF THE SUM OF DIVISORS FUNCTION

Part of: Multiplicative number theory

Published online by Cambridge University Press:  01 April 2009

JEAN-MARIE DE KONINCK  and
FLORIAN LUCA
JEAN-MARIE DE KONINCK
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec G1V OA6, Canada (email: jmdk@mat.ulaval.ca)
FLORIAN LUCA*
Affiliation:
Mathematical Institute, UNAM, Ap. Postal 61-3 (Xangari), CP 58 089, Morelia, Michoacán, Mexico (email: fluca@matmor.unam.mx)
*
✉For correspondence; e-mail: fluca@matmor.unam.mx
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given an integer n≥2, let λ(n):=(log n)/(log γ(n)), where γ(n)=∏ p∣np, denote the index of composition of n, with λ(1)=1. Letting ϕ and σ stand for the Euler function and the sum of divisors function, we show that both λ(ϕ(n)) and λ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integer k≥2, we then study the size of min {λ(ϕ(n)),λ(ϕ(n+1)),…,λ(ϕ(n+k−1))} and of min {λ(σ(n)),λ(σ(n+1)),…,λ(σ(n+k−1))} as n becomes large.

Keywords

Euler functionsum of divisors function

MSC classification

Secondary: 11N25: Distribution of integers with specified multiplicative constraints

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 2 , April 2009 , pp. 155 - 167
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The first author was supported in part by a grant from NSERC. The second author was supported in part by Grants SEP-CONACyT 46755, PAPIIT IN104505 and a Guggenheim Fellowship.

References

[1]Bassily, N. L., Kátai, I. and Wijsmuller, M., ‘On the prime power divisors of the iterates of the Euler-ϕ function’, Publ. Math. Debrecen 55(1–2) (1999), 17–32.CrossRefGoogle Scholar
[2]Browkin, J., ‘The abc-conjecture’, in: Number Theory, Trends in Mathematics (Birkhäuser, Basel, 2000), pp. 75–105.Google Scholar
[3]De Koninck, J. M. and Doyon, N., ‘À propos de l’indice de composition des nombres’, Monatsh. Math. 139 (2003), 151–167.CrossRefGoogle Scholar
[4]De Koninck, J. M. and Kátai, I., ‘On the mean value of the index of composition’, Monatsh. Math. 145 (2005), 131–144.CrossRefGoogle Scholar
[5]De Koninck, J. M., Kátai, I. and Subbarao, M. V., ‘On the index of composition of integers from various sets’, Arch. Math. 88 (2007), 524–536.CrossRefGoogle Scholar
[6]Elkies, N., ‘ABC implies Mordell’, Internat. Math. Res. Notices 7 (1991), 99–109.CrossRefGoogle Scholar
[7]Erdős, P., ‘On the normal number of prime factors of p−1 and some other related problems concerning Euler’s ϕ function’, Quart. J. Math. 6 (1935), 205–213.CrossRefGoogle Scholar
[8]Granville, A., ‘ABC allows us to count squarefrees’, Internat. Math. Res. Notices 19 (1998), 991–1009.CrossRefGoogle Scholar
[9]Ivić, A., The Riemann Zeta-Function. Theory and Applications (Dover, Minneola, NY, 2003).Google Scholar
[10]Langevin, M., ‘Partie sans facteur carré d’un produit d’entiers voisins’, in: Approximations diophantiennes et nombres transcendants (Luminy, 1990) (de Gruyter, Berlin, 1992),pp. 203–214.Google Scholar
[11]Luca, F. and Pomerance, C., ‘Irreducible radical extensions and Euler function chains’, in: Combinatorial Number Theory (Proceedings of the ‘Integers Conference 2005’ in Celebration of the 70th Birthday of R. Graham) (eds. B. Landman, M. Nathanson, J. Nešetril, R. Nowakowski and C. Pomerance) (de Gruyter, Berlin, 2007), pp. 351–362.Google Scholar
[12]Zhai, W., ‘On the mean value of the index of composition of an integer’, Acta Arith. 125 (2006), 331–345.CrossRefGoogle Scholar