Questions tagged [perfect-numbers]

Questions about or involving perfect numbers which are positive integers that are equal to the sum of their proper positive divisors.

271 questions
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Is there some sort of classification of immaculate graphs?

Suppose $\Gamma_1(V_1, E_1)$ and $\Gamma_2(V_2, E_2)$ are two finite undirected simple graphs ($V_1 \cap V_2 = \emptyset$, $E_1 \cap E_2 = \emptyset$). Let’s call $\Gamma_1 + \Gamma_2$ a graph, whose ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, then $n^2 - q^k$ is not a square.

Problem Statement Prove the following proposition. If $q^k n^2$ is an odd perfect number with special prime $q$, then $n^2 - q^k$ is not a square. Motivation Let $q^k n^2$ be an odd perfect ...
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Median divisor of even perfect numbers

I noticed that when divisors of even perfect numbers are listed in ascending order, the middle divisor (I guess the median), is always of the form $2^n$, some power of 2. If true is there a proof for ...
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On some constrained partitions of $k$-multiperfect numbers

Let $n$ be a $k$ -multiperfect number. Do we know an upper bound for the number of partitions of $n$ whose all summands are at the same time multiples of $k$ and the sum of distinct ...
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Can this inequality regarding odd perfect numbers be improved?

Let $\sigma(x)$ denote the sum of the divisors of $x$. Denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$. If $N$ is odd and $\sigma(N)=2N$, then $N$ is called an odd perfect number. Euler showed ...
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On some inequalities relating the special/Euler prime and non-Euler part of odd perfect numbers

Let $N$ be an odd (positive) integer. If $\sigma(N)=2N$ where $\sigma(N)$ is the sum of the divisors of $N$, then $N$ is called an odd perfect number. Let $I(N)=\sigma(N)/N$ denote the abundancy ...
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On Basak's “Bounds On Factors Of Odd Perfect Numbers”

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, we denote the ...
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Does mathlove's answer imply that $D(n^2) \mid n^2$?

In what follows, set $\sigma(x)$ to be the sum of divisors of $x \in \mathbb{N}$, and let $$D(x) = 2x - \sigma(x)$$ be the deficiency of $x$, and let $$s(x) = \sigma(x) - x$$ be the sum of the aliquot ...
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Why are perfect numbers called perfect numbers?

A perfect number is a number than can be expressed as a sum of its factors. For example, 28 = 1 + 2 + 4 + 7 + 14 Why is this property important? What is so perfect about perfect numbers?
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What is the common (and simplified) value for $D(q^k)D(n^2) = 2s(q^k)s(n^2)$ when $q^k n^2$ is an odd perfect number?

In an answer to an earlier question, it is shown that $$D(2^p - 1)D(2^{p-1}) = 2s(2^p - 1)s(2^{p-1}) = 2^p - 2,$$ if $2^{p-1}(2^p - 1)$ is an even perfect number, $D(x) = 2x - \sigma(x)$ is the ...
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Is $(q^k n^2 \text{ is perfect }) \iff (D(q^k)D(n^2) = 2s(q^k)s(n^2))$ only true for odd perfect numbers $q^k n^2$?

(Preamble: This question is an offshoot of this earlier MSE post.) The title says it all. Is $\bigg(q^k n^2 \text{ is perfect }\bigg) \iff \bigg(D(q^k)D(n^2) = 2s(q^k)s(n^2)\bigg)$ only true for ...
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Specific evidence for or against the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd ...
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Is $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ a square number for integer $x>1$

Motivated by this question. How to prove that $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ is not a square number for integer $x>1$? Thanks for any suggestions. Edition by the ...
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Is ${n^2}/D(n^2) \in \mathbb{N}$, if $q^k n^2$ is an odd perfect number?

Let $x \in \mathbb{N}$, the set of positive integers. The sum of the divisors of $x$ is denoted by $\sigma(x)$. Denote the deficiency of $x$ by $D(x):=2x-\sigma(x)$, and the sum of the aliquot parts ...