Displaying similar documents to “On the composition of the Euler function and the sum of divisors function”

On a divisibility problem

Horst Alzer, József Sándor (2013)

Rendiconti del Seminario Matematico della Università di Padova

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On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ

Florian Luca, Carl Pomerance (2002)

Colloquium Mathematicae

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Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n))...

The range of the sum-of-proper-divisors function

Florian Luca, Carl Pomerance (2015)

Acta Arithmetica

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Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.

Nonaliquots and Robbins numbers

William D. Banks, Florian Luca (2005)

Colloquium Mathematicae

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Let φ(·) and σ(·) denote the Euler function and the sum of divisors function, respectively. We give a lower bound for the number of m ≤ x for which the equation m = σ(n) - n has no solution. We also show that the set of positive integers m not of the form (p-1)/2 - φ(p-1) for some prime number p has a positive lower asymptotic density.

On consecutive integers divisible by the number of their divisors

Titu Andreescu, Florian Luca, M. Tip Phaovibul (2016)

Acta Arithmetica

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We prove that there are no strings of three consecutive integers each divisible by the number of its divisors, and we give an estimate for the number of positive integers n ≤ x such that each of n and n + 1 is a multiple of the number of its divisors.

On the average of the sum-of-a-divisors function

Shi-Chao Chen, Yong-Gao Chen (2004)

Colloquium Mathematicae

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We prove an Ω result on the average of the sum of the divisors of n which are relatively coprime to any given integer a. This generalizes the earlier result for a prime proved by Adhikari, Coppola and Mukhopadhyay.

Some finite generalizations of Euler's pentagonal number theorem

Ji-Cai Liu (2017)

Czechoslovak Mathematical Journal

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Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem.

On fluctuations in the mean of a sum-of-divisors function, II

Y.-F. S. Pétermann (2007)

Colloquium Mathematicae

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I give explicit values for the constant implied by an Omega-estimate due to Chen and Chen [CC] on the average of the sum of the divisors of n which are relatively coprime to any given integer a.