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On an arithmetic function considered by Pillai

Florian Luca, Ravindranathan Thangadurai (2009)

Journal de Théorie des Nombres de Bordeaux

For every positive integer n let p ( n ) be the largest prime number p ≤ n . Given a positive integer n = n 1 , we study the positive integer r = R ( n ) such that if we define recursively n i + 1 = n i - p ( n i ) for i ≥ 1 , then n r is a prime or 1 . We obtain upper bounds for R ( n ) as well as an estimate for the set of n whose R ( n ) takes on a fixed value k .

On Gelfond’s conjecture about the sum of digits of prime numbers

Joël Rivat (2009)

Journal de Théorie des Nombres de Bordeaux

The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

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

Florian Luca, Carl Pomerance (2002)

Colloquium Mathematicae

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)) < ϕ(n)...

On the counting function for the generalized Niven numbers

Ryan Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño (2009)

Journal de Théorie des Nombres de Bordeaux

Given an integer base q ≥ 2 and a completely q -additive arithmetic function f taking integer values, we deduce an asymptotic expression for the counting function N f ( x ) = # 0 ≤ n &lt; x | f ( n ) ∣ n under a mild restriction on the values of f . When f = s q , the base q sum of digits function, the integers counted by N f are the so-called base q Niven numbers, and our result provides a generalization of the asymptotic known in that case.

On the largest prime factor of n ! + 2 n - 1

Florian Luca, Igor E. Shparlinski (2005)

Journal de Théorie des Nombres de Bordeaux

For an integer n ≥ 2 we denote by P ( n ) the largest prime factor of n . We obtain several upper bounds on the number of solutions of congruences of the form n ! + 2 n - 1 ≡ 0 ( mod q ) and use these bounds to show that lim sup n → ∞ P ( n ! + 2 n - 1 ) / n ≥ ( 2 π 2 + 3 ) / 18 .

Currently displaying 21 – 40 of 58