On the mean value of the index of composition of an integer

Zhai, Wei

doi:10.4064/aa125-4-3

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction2

Heuristics1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2009

2018

Publication Types

Select...

Article5

Relationship

Self Cite2

Independent3

Authors

Journals

Cited by 5 publications

(3 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which compared with inequality (12) gives us an upper bound on n. This completes the proof of the theorem.…”

Section: Heuristicssupporting

confidence: 60%

See 1 more Smart Citation

On the Index of Composition of the Euler Function and of the Sum of Divisors Function

Koninck¹,

Luca²

2009

J. Aust. Math. Soc.

View full text Add to dashboard Cite

Given an integer n ≥ 2, let λ(n) := (log n)/(log γ (n)), where γ (n) = p|n p, 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.2000 Mathematics subject classification: primary 11N25; secondary 11A25.

show abstract

“…which compared with inequality (12) gives us an upper bound on n. This completes the proof of the theorem.…”

Section: Heuristicssupporting

confidence: 60%

“…More recently, De Koninck and Kátai [4] as well as De Koninck et al [5] have studied the distribution function of (λ(n) − 1) log n as n runs through particular sets of integers, such as the shifted primes. The mean value of the function λ(n) was also studied by Zhai [12].…”

Section: Introductionmentioning

confidence: 99%

On the Index of Composition of the Euler Function and of the Sum of Divisors Function

Koninck¹,

Luca²

2009

J. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The second-named author [7] substantially improved the results of De Koninck and Kátai via a different approach. For any fixed integer k 1, he proved the asymptotic formulas…”

Section: Introduction and Definitionsmentioning

confidence: 59%