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

Florian Luca; Carl Pomerance

Displaying similar documents to “The range of the sum-of-proper-divisors function”

On consecutive integers divisible by the number of their divisors

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.

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

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

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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.

On fluctuations in the mean of a sum-of-divisors function

Y.-F. S. Pétermann (2004)

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Triangular numbers whose sum of divisors is also triangular

Douglas E. Iannucci, Florian Luca (2007)

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

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.

On square values of the product of the Euler totient and sum of divisors functions

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.

On the average of the sum-of-p-prime-divisors function

S. D. Adhikari, G. Coppola, Anirban Mukhopadhyay (2002)

Acta Arithmetica

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Some distribution problems concerning the divisors of integers

P. Erdös, R. Hall (1974)

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Cesàro means related to the square of the divisor function

Gintautas Bareikis, Algirdas Mačiulis (2012)

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Divisors of the Euler and Carmichael functions

Kevin Ford, Yong Hu (2008)

Acta Arithmetica

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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.

On the distribution of some integers related to perfect and amicable numbers

Paul Pollack, Carl Pomerance (2013)

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The number of k-free divisors of an integer

D. Suryanarayana, V. Siva Rama Prasad (1971)

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Number of prime divisors in a product of consecutive integers

Shanta Laishram, T. N. Shorey (2004)

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Continued fractions of period five and real quadratic fields of class number one

R. Mollin, H. Williams (1990)

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On near-perfect numbers

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.