Displaying similar documents to “On the third iterates of the φ- and σ-functions”

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

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.

Lower Bounds on the Directed Sweepwidth of Planar Shapes

Markov, Minko, Haralampiev, Vladislav, Georgiev, Georgi (2015)

Serdica Journal of Computing

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We investigate a recently introduced width measure of planar shapes called sweepwidth and prove a lower bound theorem on the sweepwidth.

On a divisibility problem

Horst Alzer, József Sándor (2013)

Rendiconti del Seminario Matematico della Università di Padova

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

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.