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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.

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Relations between composite numbers, prime numbers, and perfect numbers.

(1) A composite number a is a positive integer number that is greater than 1 and can be expressed as the product of two smaller positive integer numbers, say b and c. This definition restricts b and c ...
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99 views

Number in binary as a product

I agree that any binary number that consists of $n$ ones (and no zeros) has as its decimal equivalent the number $2^n - 1$. However, the author of the book I'm reading next makes the following claim, ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, is it true that $\sigma(n^2)/q^k \mid n^2$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Question If $q^k n^2$ is an odd perfect number with special prime $q$, ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, then can $n^2 - q^k$ be a cube?

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$. This question is an offshoot of the following earlier post: If $q^k n^2$ is an odd perfect ...
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Is it possible to simplify this expression even further?

(Preamble: This question is tangentially related to this earlier one.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $...
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Finding Perfect numbers which are sum of consecutive prime number.

Find all Perfect numbers which are sum of consecutive prime number. For ex . we can write $28$ as : $$28 = 2+3+5+7+11$$ Are there any more examples possible ? If yes , what is the general condition ...
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Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function

Motivation Let $x, y$ and $z$ be positive integers. Denote the sum of divisors of $x$ by $\sigma(x)$. Also, denote the deficiency of $y$ by $D(y)=2y-\sigma(y)$. Lastly, denote the abundancy index ...
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Are there any odd perfect numbers in base $g$, where $g \neq 10$?

The topic of odd perfect numbers likely needs no introduction. Here is my primary question: Are there any odd perfect numbers in base $g$, where $g \neq 10$? Of course, there are several ...
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An interesting identity involving the abundancy index of divisors of odd perfect numbers

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $y$ is said to be perfect if $\sigma(y)=2y$. Denote the abundancy index of $z$ by $I(z)=\sigma(z)/z$. Euler proved ...
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Can the following bounds on $I(n^2) - I(q^k)$ be improved, where $q^k n^2$ is an odd perfect number with special prime $q$?

The present question is tangentially related to this earlier question of mine. Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. If $\sigma(y)=2y$, then $y$ is called a perfect ...
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64 views

Conjecture That The Only Nearly Perfect Numbers Are The Powers Of 2

Call a natural number $n$ “flawed” if its aliquot sum (the sum of its factors less than itself) is $n-1$. Show that the set of flawed numbers is equivalent to the set of perfect powers of $2$. ...
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If $q^k n^2$ is an odd perfect number with Euler prime $q$, does $I(n^2) \geq 5/3$ imply $k=1$?

Let $\sigma(x)$ denote the sum of the divisors of $x \in \mathbb{N}$. Denote the abundancy index of $y \in \mathbb{N}$ by $I(y)=\sigma(y)/y$. If $\sigma(N)=2N$, then $N$ is said to be perfect. ...
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23 views

What is the main obstruction to proving that $\omega(n) \leq k$, if $n$ is an odd deficient-perfect number and $k$ is a constant?

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. (Also, let $\omega(x)$ denote the number of distinct prime factors of $x$.) A number $n$ is called deficient-perfect if $(2n-\...
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66 views

What makes it so difficult to prove that $3$ does not divide an odd perfect number? [closed]

Let $\sigma(x)$ denote the sum of divisors the positive integer $x$. The abundancy index of $x$ is then given by the formula $I(x)=\sigma(x)/x$. A number $N \in \mathbb{N}$ is said to be perfect ...
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1answer
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Question on Potential Relationship between Mertens Function and Perfect Number Counting Function

This question assumes the following definitions where $M(x)$ is the Mertens function and $f(x)$ is the perfect number counting function. See here for more information on the Wolfram Language ...
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What is wrong with my proof of there is no odd perfect number?

The Conjecture: $A \equiv_2 1$ Implies $\sigma(A) \not = 2A$ The Proof: Let $H :=$ $[A = 2Q + 1]$ $\land$ $[\sigma(A) = 2A]$. So $H$ is true IFF $$\sigma(2Q + 1) = 2(2Q + 1).$$ Therefore we will ...
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1answer
19 views

Does $\gcd(r, \sigma(r)) = 2r - \sigma(r)$ always hold when $r$ is deficient-perfect? [duplicate]

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $r$ is called deficient-perfect if $(2r - \sigma(r)) \mid r$. Here is my question: Does $\gcd(r, \sigma(r)) = 2r - ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, then does $(q+1)/2 \mid \sigma(n^2)$ hold?

Let $\sigma(M)$ denote the sum of divisors of the positive integer $M$. If $N$ is odd and $\sigma(N)=2N$, then $N$ is called an odd perfect number. Euler showed that an odd perfect number, if one ...
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1answer
110 views

On prime factors of odd perfect numbers

Why is it so difficult to determine actual numerical values for prime factors of odd perfect numbers? Recall that, if $N$ is an odd perfect number, then Euler proved that it takes the form $N = q^k n^...
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1answer
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Does $2r - \sigma(r)$ divide $\sigma(r)$ if $r$ is deficient-perfect?

The present question is tangentially related to this earlier one. My question here is: Does $2r - \sigma(r)$ divide $\sigma(r)$ if $r$ is deficient-perfect? Recall that a positive integer $x$ is ...
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Is there a specific equation relating $\varphi(N)$ to $\sigma(N)$ when $N = q^k n^2$ is an odd perfect number?

I get $$\gcd(n^2,\sigma(n^2)) = \frac{n^2}{\sigma(q^k)/2} = \frac{\sigma(n^2)}{q^k} = \frac{2n^2 - \sigma(n^2)}{\sigma(q^{k-1})}$$ when $N = q^k n^2$ is an odd perfect number with special/Euler prime $...
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1answer
52 views

Are there (restricted) instances when the deficiency and sum-of-proper-divisors functions are multiplicative?

A function $f : \mathbb{N} \rightarrow \mathbb{Q}$ is said to be multiplicative if $$f(ab) = f(a)f(b)$$ whenever $\gcd(a,b)=1$. It is known that the sum-of-divisors function $$\sigma(x) = \sum_{d \...
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Question about a result on odd perfect numbers

In the paper titled Improving the Chen and Chen result for odd perfect numbers (Lemma 8, page 7), Broughan et al. show that if $$\frac{\sigma(n^2)}{q^k}$$ is a square, where $\sigma(x)$ is the sum of ...
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1answer
77 views

Is it possible to improve on the bounds for $\varphi(N)/N$, if $N = q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form. That is, $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. From a comment underneath this ...
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1answer
62 views

Did Descartes and Frénicle consider deficient-perfect numbers?

Holdener and Rachfal show that, if $N = q^k n^2$ is an odd perfect number with special prime $q$ (satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$), then $q^{(k-1)/2} n^2$ is deficient-...
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Does there exist a finite group that is both perfect and immaculate?

A group $G$ is called perfect iff $G’ = G$. A finite group $G$ is called immaculate iff its order is equal to the sum of orders of its proper normal subgroups. Does there exist a finite group $G$, ...
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Perfect numbers of the form $2^x 3^y$ , is my proof correct?

Prove that the only perfect number of the form $2^x 3^y$ is $6$ My proof A number $n$ is perfect if and only if: $$\sigma(n)=2n$$ $$\sigma(2^x 3^y)=2^{x+1}3^y$$ $$(2^{x+1}-1)(3^{y+1}-1)=2^{x+2}3^{y}...
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1answer
67 views

Is the Euler prime of an odd perfect number a repunit, or otherwise?

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$). (That is, $2N=\sigma(N)$ where $\sigma$ is the ...
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20 views

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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Does this iterated sequence always end in a finite number of steps to a number which is divisible by a perfect number?

I posted this question at MathOverflow, but then I realized that maybe it is more appropriate to ask it here: Let $f$ be a multiplicative arithmetic function which maps $\mathbb{N}$ to itself, such ...
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95 views

Find all prime numbers $p$ such that $5^p+ 4p^4$ is a perfect square

Find all prime numbers $p$ such that $5^p+ 4p^4$ is a perfect square.
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Odd Perfect number does not exist: where is the error in this proof

I am a math aficionado unable to find out where is the error in this proof showing that a odd perfect number does not exist. May I get some help? Suppose $X$ is a perfect number and odd. At most, $X$...
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1answer
42 views

If $N$ is deficient-perfect, under what conditions does this inequality hold?

This question is an offshoot of the following answer to a closely related MSE question. Let $N$ be a deficient-perfect number, i.e. $N$ is a positive integer such that $D(N) \mid N$ where $D(N)=2N-\...
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2answers
62 views

Can these bounds in terms of the abundancy index and deficiency functions be improved for deficient-perfect numbers?

Let $$\sigma(x) = \sum_{e \mid x}{e}$$ denote the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$, and the deficiency of $x$ by $D(x)=2x-\sigma(x)$...
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2answers
157 views

On Yanqi Xu's 2016 joint undergraduate math research project with Dr. Judy Holdener at Kenyon College

In what follows, we let $\sigma(X)$ denote the sum of the divisors of the positive integer $X$. Denote the abundancy index of $X$ by $I(X)=\sigma(X)/X$, and the deficiency of $X$ by $D(X)=2X-\sigma(X)...
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1answer
88 views

On the quantity $\sigma(\frac{n^2 \sigma(n^2)}{D(n^2)})$ when $q n^2$ is an odd perfect number with special prime $q$

Denote the sum of the divisors of $x \in \mathbb{N}$ by $\sigma(x)$. Also, denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$. If $m$ is odd and $\sigma(m)=2m$, then $m$ is called an odd perfect ...
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2answers
60 views

Breaking the barriers at $q=5$ and $q=13$ for $q^k n^2$ an odd perfect number with special prime $q$

Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. If $\sigma(N)=2N$ and $N$ is odd, then $N$ is called an odd perfect number. The question of existence of odd perfect numbers is the ...
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$n$-th term of the series 1 27 125 1000

What will be the nth term of the series 1 27 125 1000 for $n = 1$, it is 1 for $n = 2$, it is 27 for $n = 3$, it is 125 for $n = 4$, it is 1000
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Improving some known bounds regarding the abundancy index of divisors of odd perfect numbers

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the sum of aliquot ...
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1answer
54 views

Why aren't these equivalent definitions of a perfect number contradictory?

I was reading the definition of what it means for a number to be perfect and I'm a little confused. from Wikipedia: a perfect number is a positive integer that is equal to the sum of its proper ...
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On GCDs and odd perfect numbers

Let $N=q^k n^2$ be an odd perfect number with special prime $q$. The index $i(q)$ of $N$ at the prime $q$ is then equal to $$i(q):=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/2}=\frac{D(n^2)}{s(q^...
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2answers
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On the golden ratio and odd perfect numbers

Here is my question: Is $I(n^2) - 1 > 1/I(n^2)$ true when $I(n^2)=\sigma(n^2)/n^2$ is the abundancy index of $n^2$ and $q^k n^2$ is an odd perfect number with special prime $q$ satisfying $k>...
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1answer
57 views

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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115 views

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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1answer
88 views

On the Descartes-Frenicle-Sorli conjecture and the Euler prime of odd perfect numbers

(Preamble: My apologies for the somewhat long post - I merely wanted to include all the details that I had in mind for ease of reference later.) This post is an offshoot of this earlier MSE question. ...
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27 views

A consequence of assuming the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Let $x$ be a positive integer. Denote the sum of divisors of $x$ by $$\sigma(x) = \sum_{d \mid x}{d},$$ and the deficiency of $x$ by $$D(x) = 2x - \sigma(x).$$ A number $N$ is said to be perfect if $...
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4answers
166 views

For every odd $n\in\mathbb{N}$, is it true that $\sigma(n) < 2n$?

Is the following proposition true? Let $n \in \mathbb{N}$ be an odd number, then $\sigma(n) < 2n$ . For $n=p_1^\alpha p_2^\beta$ it is true : $$\sigma(n)=‎‎\left(\frac{p_1^{\alpha+1}-1}{...
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1answer
27 views

Improving the bound for $\sigma(q^k)/q^k$ where $q^k n^2$ is an odd perfect number given in Eulerian form

Let $x$ be a positive integer. (That is, let $x \in \mathbb{N}$.) We denote the sum of divisors of $x$ as $$\sigma(x) = \sum_{d \mid x}{d}.$$ We also denote the abundancy index of $x$ as $I(x)=\...
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95 views

Could a Fermat prime divide an odd perfect number?

This question is a follow-up to this earlier post. A positive integer $N$ is said to be perfect if $\sigma(N)=2N$, where $\sigma(x)$ is the sum of divisors of $x \in \mathbb{N}$. If $M$ is odd and $\...
3
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0answers
71 views

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 ...