Displaying similar documents to “Bounds for the density of abundant integers.”

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 relations between f -density and ( R ) -density

Václav Kijonka (2007)

Acta Mathematica Universitatis Ostraviensis

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In this paper it is discus a relation between f -density and ( R ) -density. A generalization of Šalát’s result concerning this relation in the case of asymptotic density is proved.

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 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))...