Displaying similar documents to “On square values of the product of the Euler totient and sum of divisors functions”

On the composition of the Euler function and the sum of divisors function

Jean-Marie De Koninck, Florian Luca (2007)

Colloquium Mathematicae

Similarity:

Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.

On a divisibility problem

Horst Alzer, József Sándor (2013)

Rendiconti del Seminario Matematico della Università di Padova

Similarity:

On consecutive integers divisible by the number of their divisors

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

Acta Arithmetica

Similarity:

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.

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

Florian Luca, Carl Pomerance (2015)

Acta Arithmetica

Similarity:

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.

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

Florian Luca, Carl Pomerance (2002)

Colloquium Mathematicae

Similarity:

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))...

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

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

Colloquium Mathematicae

Similarity:

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.

Nonaliquots and Robbins numbers

William D. Banks, Florian Luca (2005)

Colloquium Mathematicae

Similarity:

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

Y.-F. S. Pétermann (2007)

Colloquium Mathematicae

Similarity:

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.

Some finite generalizations of Euler's pentagonal number theorem

Ji-Cai Liu (2017)

Czechoslovak Mathematical Journal

Similarity:

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.