On fluctuations in the mean of a sum-of-divisors function
Y.-F. S. Pétermann (2004)
Acta Arithmetica
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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.
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.
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.
S. D. Adhikari, G. Coppola, Anirban Mukhopadhyay (2002)
Acta Arithmetica
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Douglas E. Iannucci, Florian Luca (2007)
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.
P. Erdös, R. Hall (1974)
Acta Arithmetica
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Gintautas Bareikis, Algirdas Mačiulis (2012)
Acta Arithmetica
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R. Mollin, H. Williams (1990)
Acta Arithmetica
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D. Suryanarayana, V. Siva Rama Prasad (1971)
Acta Arithmetica
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Paul Pollack, Carl Pomerance (2013)
Colloquium Mathematicae
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Milosav M. Marjanović (2005)
The Teaching of Mathematics
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Shanta Laishram, T. N. Shorey (2004)
Acta Arithmetica
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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.
Kevin Ford, Yong Hu (2008)
Acta Arithmetica
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Fernando Serrano (1995)
Journal für die reine und angewandte Mathematik
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Pascual Cutillas Ripoll (1989)
Publicacions Matemàtiques
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Let ν be a compact Riemann surface and ν' be the complement in ν of a nonvoid finite subset. Let M(ν') be the field of meromorphic functions in ν'. In this paper we study the ramification divisors of the functions in M(ν') which have exponential singularities of finite degree at the points of ν-ν', and one proves, for instance, that if a function in M(ν') belongs to the subfield generated by the functions of this type, and has a finite ramification divisor, it also has a finite divisor....
L. Hajdu, N. Saradha (2010)
Acta Arithmetica
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Min Tang, Xiaoyan Ma, Min Feng (2016)
Colloquium Mathematicae
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For a positive integer n, let σ(n) denote the sum of the positive divisors of n. We call n a near-perfect number if σ(n) = 2n + d where d is a proper divisor of n. We show that the only odd near-perfect number with four distinct prime divisors is 3⁴·7²·11²·19².