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2009•Journal of the Australian Mathematical SocietyOpen access

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

Jean–Marie De Koninck, Florian Luca

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Abstract

Abstract Given an integern≥2, letλ(n):=(logn)/(logγ(n)), whereγ(n)=∏p∣np, denote the index of composition ofn, withλ(1)=1. Letting ϕ andσstand for the Euler function and the sum of divisors function, we show that bothλ(ϕ(n)) andλ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integerk≥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))} asnbecomes large.

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Abstract Given an integern≥2, letλ(n):=(logn)/(logγ(n)), whereγ(n)=∏p∣np, denote the index of composition ofn, withλ(1)=1. Letting ϕ andσstand for the Euler function and the sum of divisors function, we show that bothλ(ϕ(n)) andλ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integerk≥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))} asnbecomes large.

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Abstract Given an integern≥2, letλ(n):=(logn)/(logγ(n)), whereγ(n)=∏p∣np, denote the index of composition ofn, withλ(1)=1. Letting ϕ andσstand for the Euler function and the sum of divisors function, we show that bothλ(ϕ(n)) andλ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integerk≥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))} asnbecomes large.

Key concepts: Mathematics, Integer (computer science), Composition (language), Function (biology), Combinatorics, Order (exchange), Value (mathematics), Divisor function

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