Divisors of the Euler and Carmichael functions
Kevin Ford, Yong Hu (2008)
Acta Arithmetica
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Kevin Ford, Yong Hu (2008)
Acta Arithmetica
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Kevin Broughan, Kevin Ford, Florian Luca (2013)
Colloquium Mathematicae
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If n is a positive integer such that ϕ(n)σ(n) = m² for some positive integer m, then m ≤ n. We put m = n-a and we study the positive integers a arising in this way.
Jean-Marie De Koninck, Florian Luca (2007)
Colloquium Mathematicae
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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.
Y.-F. S. Pétermann (2004)
Acta Arithmetica
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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.
Horst Alzer, József Sándor (2013)
Rendiconti del Seminario Matematico della Università di Padova
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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.
Florian Luca, Carl Pomerance (2012)
Colloquium Mathematicae
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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.
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.
S. D. Adhikari, G. Coppola, Anirban Mukhopadhyay (2002)
Acta Arithmetica
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P. Erdös, R. Hall (1974)
Acta Arithmetica
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Gintautas Bareikis, Algirdas Mačiulis (2012)
Acta Arithmetica
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W. Narkiewicz (1981)
Journal für die reine und angewandte Mathematik
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Mohand-Ouamar Hernane, Florian Luca (2009)
Acta Arithmetica
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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))...
L. Hajdu, N. Saradha (2010)
Acta Arithmetica
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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.
William D. Banks, Igor E. Shparlinski (2004)
Acta Arithmetica
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D. Suryanarayana, V. Siva Rama Prasad (1971)
Acta Arithmetica
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Fernando Serrano (1995)
Journal für die reine und angewandte Mathematik
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R. Mollin, H. Williams (1990)
Acta Arithmetica
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