Results 11 to 20 of about 41,624 (236)

On the number of prime factors of values of the sum-of-proper-divisors function [PDF]

Journal of Number Theory, 2015
Let $\omega(n)$ (resp. $\Omega(n)$) denote the number of prime divisors (resp. with multiplicity) of a natural number $n$. In 1917, Hardy and Ramanujan proved that the normal order of $\omega(n)$ is $\log\log n$, and the same is true of $\Omega(n ...
Troupe, Lee
core   +2 more sources

Practical numbers and the distribution of divisors

Quarterly Journal of Mathematics, 2015
An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern.
Weingartner, Andreas
core   +3 more sources

From sums of divisors to partition congruences

Revista De La Real Academia De Ciencias Exactas, Fisicas Y Naturales - Serie A: Matematicas
Let z be a complex number. For any positive integer n it is well known that the sum of the zth powers of the positive divisors of n can be computed without knowing all the divisors of n, if we take into account the factorization of n.
M. Merca
semanticscholar   +3 more sources

Iterating the Sum-of-Divisors Function [PDF]

Experimental Mathematics, 1996
Let $\sigma^0(n) = n$ and $\sigma^m(n) = \sigma(\sigma^{m-1}(n))$, where $m\ge1$ and $\sigma$ is the sum-of-divisors function. We say that $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$. We have tabulated all $(2,k)$-perfect numbers up to $10^9$ and all $(3,k)$- and $(4,k)$-perfect numbers up to $2\cdot10^8$.
G. Cohen, H. T. Riele
semanticscholar   +3 more sources

The Dirichlet divisor problem over square-free integers and unitary convolutions [PDF]

Notes on Number Theory and Discrete Mathematics, 2023
We obtain an asymptotic formula for the sum ~D₂ of the divisors of all square-free integers less than or equal to x, with error term O(x^{1/2 + ε}). This improves the error term O(x^{3/4 + ε}) presented in [7] obtained via analytical methods.
André Pierro de Camargo
doaj   +1 more source

Arithmetic properties of the sum of divisors [PDF]

Journal of Number Theory, 2020
The divisor function $\sigma(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$.
T. Amdeberhan, V. Moll, Vaishavi Sharma, Diego Villamizar   +3 more
semanticscholar   +1 more source

Recurrences for the sum of divisors [PDF]

Proceedings of the American Mathematical Society, 1977
The author presents two recursive determinations of the sum of positive divisors of a given positive integer. Each recurrence is then discussed with regard to economy of computation, and in this light is compared with the well-known recurrence of Niven and Zuckerman.
J. A. Ewell
semanticscholar   +2 more sources

Powerfree sums of proper divisors [PDF]

Colloquium Mathematicum, 2022
Let $s(n):= \sum_{d\mid n ...
Pollack, Paul, Roy, Akash Singha
openaire   +2 more sources

Dynamic multi‐objective optimisation of complex networks based on evolutionary computation

IET Networks, EarlyView., 2022
Abstract As the problems concerning the number of information to be optimised is increasing, the optimisation level is getting higher, the target information is more diversified, and the algorithms are becoming more complex; the traditional algorithms such as particle swarm and differential evolution are far from being able to deal with this situation ...
Linfeng Huang
wiley   +1 more source

On divisors of sums of integers. I [PDF]

Acta Mathematica Hungarica, 1986
Throughout this article c 0, c 1, c 2, … will denote effectively computable positive absolute constants. Denote the cardinality of a set X by |X|. Let N be a positive integer and let A and B be non-empty subsets of {1, …,N ...
Sárközy, A., Stewart, C.L.
openaire   +4 more sources