Questions tagged [perfect-numbers]
A perfect number is a positive integer that is equal to the sum of its proper positive divisors.
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On J. T. Condict's Senior Thesis on Odd Perfect Numbers
I am trying to locate a copy of J. T. Condict's senior thesis on odd perfect numbers:
J. Condict, On an odd perfect number's largest prime divisor, Senior Thesis, Middlebury College (1978).
I am ...
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Perfect numbers and perfect powers
This was asked earlier at MSE.
The observation that 28 = 27 + 1 shows that it is possible to have consecutive perfect numbers and perfect powers. However, this must be extremely rare. Is it ...
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Has it been proved that odd perfect numbers cannot be triangular?
(Note: This question has been cross-posted from MSE.)
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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Perfect Runs of Consecutive Integers
A run of 2 or more consecutive integers is said to be perfect if the sum of its terms equals the sum of all the their proper divisors, adding common divisors as often as they occur. For example, 672, ...
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Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd perfect number?
Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd perfect number?
I have asked the same question in MSE, but did not get any answers. I was wondering if ...
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Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
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Generalized quasi-perfect numbers
A number $n \in \mathbb{N}$ is called quasi-perfect if $\sigma(n) = 2n+1$, where $\sigma$ is the sum of divisors function. It is known that if $n$ is quasi-perfect, then it must be the square of an ...
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Can perfect numbers be seen $p$-adically?
It is well known that all even perfect numbers are of the form $N=(2^{q}-1).2^{q-1}$ with $M_{q}:=2^{q}-1$ a Mersenne prime.
As the very defining property of such a perfect number is to fulfill the ...
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Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?
This question is related to the last question about van der Pol's identity for the sum of divisors.
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
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An equation involving perfect numbers
Let $s,x_1,x_2,\cdots, x_s$ be natural numbers not neccesarily distinct.
I am interested in solving the equation $$(x_1+x_2+\cdots +x_s)^s=2^s(x_1\cdot x_2\cdots x_s)^2$$
Some Notes:
I have found ...
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Perfect numbers, Galois groups and a polynomial
Let $f(n,t) = \sum_{k=0}^{r-1} d_k t^k$ where $D_n = \{d_0=1,d_1,\cdots,d_{r-1}\}$ are all divisors of $n$.
For instance
$$f(28,t) = 28 t^{5} + 14 t^{4} + 7 t^{3} + 4 t^{2} + 2 t + 1$$
For even ...
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Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer? [closed]
After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for $...
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Kindda-Perfect number: Is there a sequence of numbers which are equal to the sum of its proper divisors excluding itself as well as 1?
Perfect number is a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3.
Is there a sequence of numbers which are equal ...
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Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers
I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's Equation
$$y^2 = x^3 +...
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Inductively computing Mersenne primes / perfect numbers?
For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$.
Let $(x_n)_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$.
Set $\hat{\phi}(1) = \{x_1\}$ and ...
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Conjecture on odd perfect numbers
I'm a grad student in mathematics and I've been working with a very gifted high school student (likely the smartest high school student I've ever met) on problems he's brought up and some competition ...
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Can an even perfect number be a sum of two cubes?
A similar question was asked before in https://math.stackexchange.com/questions/2727090/even-perfect-number-that-is-also-a-sum-of-two-cubes, but no conclusions were drawn.
On the Wikipedia article of ...
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Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?
I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...
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Touchard / van der Pol's identity for the sum of divisors and an elliptic curve for perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
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Integers n such that sigma(n)=omega(n)n and omega(n) divides n
Are there other integers $n$ than even perfect numbers such that $\sigma(n)=\omega(n)n$ and $\omega(n)\vert n$?
Thanks in advance.
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On Sorli's Conjecture Re: OPNs (Circa 2003)
In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" (hyperlinked here) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler ...
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What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?
For integers $n\geq 1$ I denote the Euler's totient function as $\varphi(n)$ and the divisor function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$, that are two well-known mulitplicative functions. We ...
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If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ a square?
(I have asked a similar question in MSE four days ago, but did not receive any answers. I have therefore cross-posted it to this site, hoping to get some responses.)
An odd perfect number $N$ is ...
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The action of the unitary divisors group on the set of divisors and odd perfect numbers
Let $n$ be a natural number. Let $U_n = \{d \in \mathbb{N}\mid d\mid n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D_n$ be the set of divisors and $S_n=\{d \in \mathbb{N}\mid d^2 \...
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Divisibility relation with a specific sum of divisors
Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$
I have checked this up to $n=100$, and I ...
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On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
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Frey's elliptic curve and perfect numbers?
Let $E_n:y^2=x(x-\sigma(n)/2)(x+\sigma(n)/n)$ be a Frey-elliptic curve, where $\sigma$ denotes the sum of divisors of the natural number $n$.
If $n$ is a perfect number ($\sigma(n)=2n$) then the $j$-...
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does infinite leinster groups indicates infinite perfect numbers?
The simplest definition i managed to find is: Leinster group is a finite group which her order equal the sum of its normal subgroups order, and as a perfect number is a positive integer that is equal ...
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If $N = qn^2$ is an odd perfect number with $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?
The title says it all.
Question
If $N = qn^2$ is an odd perfect number with Euler prime $q$ and $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, with ...
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Proof of the Infinitude of Odd Primitive Pseudoperfect Numbers
I'm interested in the infinitude of odd primitive pseudoperfect numbers. Richard K. Guy's book "Unsolved Problems in Number Theory 3rd edition" says that P. Erdős proved the infinitude of odd ...
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A conjecture regarding odd perfect numbers
(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of ...
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Solutions of $rad(\sigma(m))=2rad(m)$
For $m$ a positive integer greater than $1$, let $rad(m)$ be the product of all distinct primes dividing $m$. If $n$ is an odd perfect number (conjectured not to exist), one would have $\sigma(n)=2n$, ...
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Bounds for the number of prime numbers less than the Euler's factor, the radical and the greatest prime factor, respectively, of an odd perfect number
As tell us the Wikipedia section dedicated to Odd perfect numbers (please, see also the related references if you need it), any perfect number has the form $$n=q^\alpha m^2$$
where the integer $\alpha\...
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On the largest prime factor and the largest component of an odd perfect number
(1) The largest component $p^a$ of an odd perfect number is known to be greater than $10^{62}$.
(2) The largest prime of an odd perfect number is known to be greater than $10^8$.
Does (1) imply ...
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Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes
I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...
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On odd perfect numbers and a GCD
(Note: This question is closely related to this other one in MSE.)
Let $N = q^k n^2$ be an odd perfect number.
From this paper in NNTDM, we have the equation
$$i(q) := \frac{\sigma(n^2)}{q^k}=\frac{...
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Help with R. Ryan's "A simpler dense proof regarding the abundancy Index."
I'm reading Richard Ryan's article "A simpler dense proof regarding the abundancy index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows:
Suppose we have a ...
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Abelian characters and odd perfect numbers?
This question is about applications of abelian characters to odd perfect numbers:
Context and Definitions:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring ...
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Does this Theorem 2 from Dandapat et al. imply that $\gcd(\sigma(p^k),\sigma(a^2)) > 1$?
Write the odd perfect number $m=p^k a^2$ as a product of primes
$$m = p^k {p_1}^{2a_1} \cdots {p_v}^{2a_v}.$$
(Note that it is known that $v \geq 9$ by work of Nielsen.) Let $N(m)$ be the number of ...
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Looking for an appropriate reference(s) for two conjectures on odd perfect numbers
(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)
Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For ...
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Does there exist an integer that is both solitary and almost perfect?
This question is an offshoot from the following MSE post. I hope that it is appropriate for this site.
Let $\sigma(x)$ be the sum of the divisors of $x$.
An integer $a$ is said to be solitary if ...
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A possible axiomatic characterization of the set of divisors of a perfect number
Define a pro-perfect set $S$ to be a finite set of positive integers satisfying the following three properties:
$1\in S$.
$\displaystyle\sum_{n\in S}n^{-1}\in S$
There exists a unique permutation $\...
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Any results towards the irrationality of the sum of reciprocals of perfect numbers? [closed]
This question is a follow up to my comment to Sum of the reciprocal of perfect numbers. I would like to know which results have been published about the possible irrationality of the sum of ...
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A Problem Concerning Odd Perfect Number
Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number.
I know there are results much stronger than the one above, but I am looking for an answer ...
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On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number
This question has been cross-posted from this MSE question and is an offshoot of this other MSE question.
(Note that MSE user mathlove has posted an answer in MSE, which I could not completely ...
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II
(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.)
The topic of odd perfect ...
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Is $\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$ finite for every $k$?
I would like to check if this limit :$$\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$$ finite for every $k$?
where :$\phi_{k}$ is iterating Euler - totient function ...
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On odd perfect numbers $q^k n^2$ satisfying $n^2 - q^k = 2^r t$
Let $N = q^k n^2$ be an odd perfect number with special prime $q$, satisfying
$$n^2 - q^k = 2^r t$$
where $r \geq 2$ and $\gcd(2,t)=1$.
We could prove that:
(1) $2^r t > 2n$. (We can modestly ...
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A geometric approach to the odd perfect number problem?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function.
Define:
$$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
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On attempting a proof for $r > 1$, if $M = {2^r}{b^2}$ is an even almost perfect number which is not a power of two
(Preamble: I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that brilliant idea can help with answering my ...