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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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A conjecture regarding odd perfect numbers

(Note: This question has now been cross-posted to MO.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $z$ by $D(z):=2z-\...
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Does there exist an integer $m$ which is not in the image of $D(n)=2n - \sigma(n)$?

Let $\sigma(n)$ be the sum of the divisors of the positive integer $n$, and denote the deficiency of $n$ by $D(n)=2n-\sigma(n)$. Here is my question: Do there exist numbers $m$ which (provably) do ...
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Any value $n$ we can conclude there are no perfect numbers of the form $kn+1$ or congruent to $1$ $\pmod n$?

Is it proven for any integer $n$ that there are no perfect numbers of the form $kn+1$ or more simply it has been proven there are no perfect numbers congruent to $1$ $\pmod n$? For instance, $n$ $=$ $...
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Is the occurence of two perfect numbers a coincidence?

The expression $$n^n+n!+1$$ is prime for the following non-negative integers $n\le 7\ 600$ : $$[0, 1, 2, 4, 28, 496]$$ if we assume $0^0=1$ The numbers $28$ and $496$ are perfect numbers. Is ...
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110 views

Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted to MO.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
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91 views

Is there a notion “above” that of perfect numbers?

When trying to understand a notion, it often gives great insight to see it as the "shadow" of something bigger, carrying more information. The notion of categorification relies on this idea. A basic ...
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111 views

If we suppose, that a perfect odd number exists, can we determine whether infinite many exist?

It is currently not known whether odd perfect numbers (numbers with the property $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$, including $1$ and $n$) exist. But suppose, a perfect ...
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151 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, can $q=17$ hold?

Note: This question is an offshoot of this earlier MSE post. If $N$ is odd and $\sigma(N)=2N$ where $\sigma=\sigma_{1}$ is the classical sum-of-divisors function, then $N$ is said to be an odd ...
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77 views

If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
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84 views

A relation related with odd perfect numbers

It is easy to prove, using the relation $\prod_{d\mid n}d=n^{\sigma_0(n)/2}$ holds for $n\geq 1$ where $\sigma_0(n)$ is the number of divisors, the following Proposition. The integer $n\geq 1$ is ...
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39 views

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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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 ...
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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 $\...
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145 views

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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119 views

Odd perfect numbers, questions on a proof of Steuerwald

I'm currently working on odd perfect numbers, to be precise on a proof (german language) of Rudolf Steuerwald. I have two questions regarding this. Euler proved that any odd perfect number $n$ has to ...
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<Reference Request> Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This has been cross-posted to MO.) Good day! I would like to request for references to research done as to whether the Euler prime of an odd perfect number can also be its largest factor. ...
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127 views

Are there any Perfectly Abundant Numbers?

Define a Perfectly Abundant Number as a number with the property: $$ x : \sigma(x) = 2x + 1, $$ where $\sigma$ is the sum of the factors of $x$, including itself. In other words, $x$ isn't a perfect ...
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Let $a_n=2^{2n}\left(2^{2n+1}-1 \right)$. Show induction of $n$

Let $a_n=2^{2n}\left(2^{2n+1}-1 \right)$. Show induction of $n$ $$a_{2n+1}=256a_{2n-1}+60\left(16^n\right)$$ $$a_{2n+2}=256a_{2n}+240\left(16^n\right)$$ I tried $n=1$, $a_3=2^{2\cdot3}\...
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340 views

Can an odd perfect number be divisible by $101$?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
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1answer
87 views

Equations involving arithmetic functions, totatives and even perfect numbers

I've deduced simple relationships that satisfy each even perfecf number (even numbers $n$ for which $\sum_{d\mid n}d=2n$) and now I wondered about related conjectures. For each integer $m\geq 1$ we ...
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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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102 views

Is there an anomaly in the distribution of Mersenne primes?

In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that $$2^{82589933} - 1$$ is the largest known (Mersenne) prime, ...
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Are there any other squares $n^2$ for which $\gcd(n^2, \sigma(n^2)) = 2n^2 - \sigma(n^2)$?

Let $\sigma(x)$ denote the sum of the divisors of the positive integer $x$. Denote the deficiency of $x$ by $$D(x)=2x-\sigma(x).$$ I am interested in solutions to the equation $$\gcd(n^2, \sigma(n^2)...
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Inquiry about the Wolfram MathWorld entry on Odd Perfect Number

I just have a quick inquiry about the Wolfram MathWorld entry on Odd Perfect Number. Last sentence of the fifth paragraph states that: Hagis (1980) showed that odd perfect numbers must have at ...
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118 views

Equations involving the Euler's totient function and Mersenne primes

In this post we denote the Euler's totient function as $\varphi(n)$, first we show a claim related to Mersenne primes, see for example this Wikipedia and secondly we are going to ask a related ...
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86 views

Can the following argument be pushed to a full proof that $(p + 2)/p$ is an outlaw if $p$ is an odd prime?

This is related to this earlier MSE question. Let $\sigma(x)$ be the sum of the divisors of $x$, and denote the abundancy index of $x$ by $I(x):=\sigma(x)/x$. If the equation $I(a) = b/c$ has no ...
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103 views

Even perfect number that is also a sum of two cubes

In 2010, Gallardo proved that the only even perfect number that is also a sum of two cubes is $28$. Here is a link to his proof (see the second page). The first part goes roughly as follows: Let $N$ ...
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103 views

Does the existence of infinite number of Leinster groups indicate the existence of infinite number of perfect numbers?

A Leinster group is a finite group $G$, such, that that the sum of orders of its normal subgroups is $2|G|$. A perfect number is a positive integer that is equal to the sum of its proper positive ...
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147 views

What about the equation $\sigma(2n)=2\left(n+\sigma(n)\right),$ involving the sum of divisor function?

Yesterday I wrote this equation involving the sum of divisors function $\sigma(l)=\sum_{d\mid l}d$, $$\sigma(2n)=2\left(n+\sigma(n)\right).\tag{1}$$ Due to its very simple form I don't know if it is ...
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26 views

Are there imperfects or pluperfects numbers?

I am not a mathematician nor a cientist, I'm just a curious person. My math background is always trying to be "back there", as anything you learn tends to be. So, there is a risk that I post silly ...
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78 views

Assuming the Lenstra–Pomerance–Wagstaff conjecture, in what range of values for $p$ would we expect to find the next prime $2^p - 1$?

Assuming the Lenstra–Pomerance–Wagstaff conjecture is correct, in what range of values for a prime $p$ would we expect to find the next Mersenne prime $2^p - 1$?
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On a conjectured relationship between the least prime factor and the Euler prime of an odd perfect number

(Note: This question has been cross-posted to MO.) Let $\sigma$ be the classical sum-of-divisors function. For example, $$\sigma(6)=1+2+3+6=12={2}\cdot{6}.$$ If $\sigma(N)=2N$ and $N$ is odd, then $...
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55 views

On identities and congruences involving the harmonic mean of odd perfect numbers

I would like to know if my calculations were rights and if you want to deduce other different interesting congruences following my approach. On assumption (notice that it isn't the usually accepted ...
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Can an odd perfect number be divisible by either $2049$ or $2051$?

Can an odd perfect number be divisible by either $2049$ or $2051$? Note that $2049 = 3 \cdot {683}$, and that $2051 = 7 \cdot {293}$. Added July 15 2016 It is known that an odd perfect number ...
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A cubic relationship for perfect numbers

Combining the identity (the composition of the polynomial $P_3(x)=\frac{x(x+1)(2x+1)}{6}$ with the arithmetical function $N(n)=n$ and the known proof by induction) $$1^2+2^2+\ldots+n^2=\frac{n(n+1)(...
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2answers
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Proving this quotient is always even.

Let $n$ be an odd, composite integer. Let $D$ equals the sum of the proper divisors of $n$ minus the last divisor, $d_j$. Also, let $n$ be a number such that $D \gt \frac{1}{2} \times{n}$. Let $d_i$ ...
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29 views

Requesting for details of Grun's result on odd perfect numbers

This post is an offshoot of this earlier MSE question. According to this Wikipedia page: Grun $1952$ proved that the smallest prime factor $p_1$ (of an odd perfect number $N$) is $< \frac{2\...
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94 views

Does the Descartes number $D = 198585576189$ have a friend?

Let $\sigma(X)$ be the sum of the divisors of $X$, and denote the abundancy index $\sigma(X)/X$ by $I(X)$. If the equation $I(X) = r/s$ has no solution $X \in \mathbb{N}$, then $r/s$ is said to be an ...
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151 views

If n>6 is an even perfect number, Prove that n is congruent to 4 (mod12)

I know that for n to be an even perfect number greater than 6 it has the form $ 2^(m-1)(2^m-1)$ where m is prime. I also know that since n is an even perfect number, it is congruent to 1 (mod 9). ...
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55 views

With $rad(N)=\prod_{p|N}p$, if $N$ is even and $\frac{2+rad(N)}{8}\left(\sum_{\substack{d|N,d<rad(N)}}d\right)=N$ then is perfect?

In the literature (see for example sites and paper concerning to the abc conjecture, I say this as reference and by caution to avoid mistakes) is defined the arithmetical function $rad(n)$ as $1$ if $...
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172 views

Can an odd perfect number be divisible by $119$?

We know that an odd perfect number cannot be divisible by $105$ (see here), by $825$ (see here), and by $5313$ (see here). I wonder if that's also the case for $119$.
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188 views

On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
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121 views

What do we know about Sorli's Conjecture and Odd Perfect Numbers?

I have seen that Jose Arnaldo Dris has recently published a great deal of literature concerning odd perfect numbers and Sorli's Conjecture. He claims that the truth of Sorli's conjecture implies that ...
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Combinatorial interpretation of Euclid's form for even perfect numbers

Euclid showed that if $p$ is a prime such that $2^{p}-1$ is also a prime, then the number $n=2^{p-1}.(2^{p}-1)$ is perfect. Much later, Euler proved that every even perfect number is of this form. ...
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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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31 views

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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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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33 views

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
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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41 views

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 $...