April 7, 2015
We say a natural number~$n$ is abundant if $\sigma(n)>2n$, where $\sigma(n)$ denotes the sum of the divisors of~$n$. The aliquot parts of~$n$ are those divisors less than~$n$, and we say that an abundant number~$n$ is pseudoperfect if there is some subset of the aliquot parts of~$n$ which sum to~$n$. We say~$n$ is weird if~$n$ is abundant but not pseudoperfect. We call a weird number~$n$ primitive if none of its aliquot parts are weird. We find all primitive weird numbers of the form $2^kpq$ ($p<q$ being odd primes) for $1\le k\le14$. We also find primitive weird numbers of the same form, larger than any previously published.
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March 1, 2018
In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a given number of distinct prime factors are presented. These algorithms yield primitive weird numbers of the form $mp_1\dots p_k$ for a suitable deficient positive integer $m$ and primes $p_1,\dots,p_k$ and generalize a recent technique develope...
February 20, 2018
We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $\Omega$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $\Omega=6$, count all PANs and square-free PANs up to $\Omega=7$ and count all odd PANs and odd square-free PANs up to $\Omega=8$. We find primitive weird numbers (briefly, PWNs) with up to 16 prime factors, improving the previous results of [Amato-Hasler-Melfi-Parton] where PWN...
July 26, 2022
Weird numbers are abundant numbers that are not pseudoperfect. Since their introduction, the existence of odd weird numbers has been an open problem. In this work, we describe our computational effort to search for odd weird numbers, which shows their non-existence up to $10^{21}$. We also searched up to $10^{28}$ for numbers with an abundance below $10^{14}$, to no avail. Our approach to speed up the search can be viewed as an application of reverse search in the domain of c...
August 5, 2021
In this note, we continue an approach pursued in an earlier paper of the second author and thereby attempt to produce an improved lower bound for the sum $I(q^k) + I(n^2)$, where $q^k n^2$ is an odd perfect number with special prime $q$ and $I(x)$ is the abundancy index of the positive integer $x$. In particular, this yields an upper bound for $k$.
June 18, 2019
For a positive integer $n$, if $\sigma(n)$ denotes the sum of the positive divisors of $n$, then $n$ is called a deficient perfect number if $\sigma(n)=2n-d$ for some positive divisor $d$ of $n$. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.
August 24, 2020
The divisor function $\sigma(n)$ sums the divisors of $n$. We call $n$ abundant when $\sigma(n) - n > n$ and perfect when $\sigma(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor function. It measures the extent to which a number is highly divisible into parts, such that the parts are highly divisible into subparts, so on. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor...
June 14, 2012
If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{\sigma(p^k)}{p^k}$ and $\frac{\sigma(m^2)}{m^2}$, and the ratios $\frac{\sigma(p^k)}{m^2}$ and $\frac{\sigma(m^2)}{p^k}$, are used. It is also showed that $m^2 > \frac{\sqrt{6}}{2}({10}^{150})$.
August 14, 2019
For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma(n)=2n-d$. In this paper, we show that the only odd deficient-perfect number with four distinct prime divisors is $3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}$.
May 25, 2020
Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. We also obtain tighter bounds for odd perfect numbers. We also discuss the behavior of $\sigma(n!+1)$, $\sigma(2^n+1)$, and related sequences.
January 20, 2020
Let $n$ and $k$ be positive integers and $\sigma(n)$ the sum of all positive divisors of $n$. We call $n$ an exactly $k$-deficient-perfect number with deficient divisors $d_1, d_2, \ldots, d_k$ if $d_1, d_2, \ldots, d_k$ are distinct proper divisors of $n$ and $\sigma (n)=2n-(d_1+d_2+\ldots + d_k)$. In this article, we show that the only odd exactly $3$-deficient-perfect number with at most two distinct prime factors is $1521=3^2 \cdot 13^2$.