Displaying similar documents to “On the sumset of the primes and a linear recurrence”

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.

Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields

Hans Roskam (2001)

Journal de théorie des nombres de Bordeaux

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Let S be a linear integer recurrent sequence of order k ≥ 3 , and define P S as the set of primes that divide at least one term of S . We give a heuristic approach to the problem whether P S has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that P S has positive lower density for “generic” sequences S . Some numerical examples are included.

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))...