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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
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Abstract

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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
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.

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