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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Can $1!+2!+3!+\cdots n!$ be a perfect number?
Can $\sum_{k=1}^nk!$ be a perfect number?
I think an odd perfect number can be divisible by $9$ and the last digit of an odd perfect number can be $3$, so I think the above sum can be a perfect number....
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Revisiting MSE question 4386812 - Determining whether $\sigma(q^k)/2$ is squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$
Preamble: The present inquiry is an offshoot of this earlier MSE question.
MOTIVATION
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the ...
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Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
It is known that
$$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$
where $D(x)=2x-\...
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If $p^k m^2$ is an odd perfect number, then $D(p^k)/s(p^k)$ is in lowest terms. Does this contradict $D(p^k)D(m^2)=2s(p^k)s(m^2)$?
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect number.
Euler showed ...
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Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$.
A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer.
The number $1$ ...
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If $p^k m^2$ is an odd perfect number with special prime $p$ and $p = k$, then $\sigma(p^k)/2$ is not squarefree.
While researching the topic of odd perfect numbers, we came across the following implication, which we currently do not know how to prove:
CONJECTURE: If $p^k m^2$ is an odd perfect number with ...
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Is this logical deduction regarding some modular restrictions on odd perfect numbers valid? - Part II
Let $p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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Is this logical deduction regarding some modular restrictions on odd perfect numbers valid?
Let $p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
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If $q^k n^2$ is an odd perfect number, then $n^2 - q^k = 2^r t$ implies that $3 \leq r$ is odd. Therefore?
The topic of odd perfect numbers likely needs no introduction.
Let $N$ be an odd perfect number given in the so-called Eulerian form $N = q^k n^2$ where $q$ is the special prime satisfying $q \equiv k ...
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On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)
Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
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Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part III
(Preamble: This question is an offshoot of the following inquiry in MSE.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$...
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Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II
This MSE question (from April 2020) asked whether the inequality
$$\frac{D(n^2)}{s(n^2)} < \frac{D(n)}{s(n)}$$
could be improved, where $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer ...
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On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$
Let $N$ be an odd perfect number given in the so-called Eulerian form
$$N = q^k n^2$$
where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
In what follows, let ...
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On Tony Kuria Kimani's recent preprint in ResearchGate
(Preamble: The method presented here to compute the GCD $g$ is patterned after the method used to compute a similar GCD in this answer to a closely related MSE question.)
Let $\sigma(x)=\sigma_1(x)$ ...
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Is $P+1$ prime for the perfect number $P$ corresponding to the exponent $74207281$?
The even perfect numbers are closely related to the Mersenne primes. We currently know $51$ Mersenne primes and hence $51$ perfect numbers.
It has already been checked for which of those perfect ...
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Proof-verification request: On the equation $\gcd(n^2,\sigma(n^2))=D(n^2)/s(q^k)$ - Part II
(Preamble: This inquiry is an offshoot of this answer to a closely related question.)
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the ...
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Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II
(Preamble: This inquiry is an offshoot of this MSE question.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
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Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$.
It is known that
$$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
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Proof-verification request: On the equation $\gcd(n^2,\sigma(n^2)) = D(n^2)/s(q^k)$
(Preamble: This question is an offshoot of this earlier post.)
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(...
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Are the (Bezout) coefficients for $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1)n^2$ (where $q^k n^2$ is an odd perfect number) unique?
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the aliquot sum of $x$ by $s(x)=\sigma(x)...
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Is there any other known relationship between even perfect numbers and odd perfect numbers, apart from their multiplicative forms?
(Note: This was cross-posted from MO, because it was not well-received there. Will delete the MO post in a few.)
Observe that an even perfect number $M = (2^p - 1)\cdot{2^{p - 1}}$ and an odd perfect ...
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Is the argument used in this proof that $k=1$ logically sound, where $q^k n^2$ is an odd perfect number with special prime $q$?
The topic of odd perfect numbers likely needs no introduction.
Euler proved that a hypothetical odd perfect number $N$, if one exists, must have the so-called Eulerian form $N=q^k n^2$, where $q$ is ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, is it possible to express $\sigma(n^2)/q^k$ as a function of only $q$ and $k$?
Let $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$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?
Preamble: This question is an offshoot of this earlier post. (This inquiry has likewise been cross-posted to MO last June $10, 2022$.)
Let $N = q^k n^2$ be an odd perfect number with special prime $q$...
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Follow-up to MSE question 3738458
This is a follow-up inquiry to this MSE question.
Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$.
A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
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Primes with prime digits and prime sum of the digits
I don't know if this concept is already defined, I consider the number $n$ to be a perfect prime (weak) if $n$ is a prime, its digits are primes and the sum of the digits is prime. Every one digit ...
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Can this inequality be solved for $q$ in terms of $n$ (or the other way around), if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $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$.
The following general inequality relating $k$ and $q$ is proved in this ...
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What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?
The following query is an offshoot of this answer to a closely related post.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect ...
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On Carmichael function and aliquot parts of odd perfect numbers
This post is cross-posted on MathOverflow with identifier 439563 and same title.
We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\...
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Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?
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$.
Define the GCDs:
$$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$
$$H = \...
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Reference request regarding odd 4-perfect numbers
I am reading a paper by Broughan and Zhou (2006) that is dealing with odd 4-perfect numbers. The title of the paper is "Odd multiperfect numbers of abundancy four."
In this paper they ...
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semi perfect number - number of divisors
The definition:
we define number as semi perfect , if the number equals to the sum of exactly k of its divisors.
the question:
prove that for every n (n>0 | n belong to N)
n is semi perfect order 3 ...
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Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The following query is an offshoot of this post 1 and this post 2.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect numbers likely ...
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What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the divisor sum $\sigma$ is a multiplicative function.)
A number $P$ is said to be perfect if $\...
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I have found a proof that there does not exist any odd perfect number, Am I Correct? [closed]
To Verify the proof click here
I really cannot imagine I did it, My Intuition to send this here is just please God there must be a mistake in here, but if magically there is isn't, I will really say ...
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Perfect numbers and Pell's equation
(Disclaimer: The following is a naive attempt to apply the theory of Pell's equations to perfect numbers. Please bear in mind that this is my first time to try solving such an equation.)
Let $p^k$ be ...
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On odd perfect numbers and a GCD - Part VII
(Pardon me for being somewhat stubborn, but this question will be the last for this week. This post is an offshoot of this one.)
Let $N = q^k n^2$ be an odd perfect number be an odd perfect number ...
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Does $G \mid I$ and $I \mid H$ still hold if $\sigma(q^k)/2$ is not squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$?
This question is an offshoot of this post #1 and this post #2.
Let $N = q^k n^2$ be an odd perfect number be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\...
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Please check my proof: If $q^k n^2$ is an odd perfect number with special prime $q$, then $\sigma(q^k)/2 \mid n$ if and only if $n \mid \sigma(n^2)$.
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the classical sum of divisors of the positive integer $x$ by $\...
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Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?
This question is an offshoot of this MSE answer.
Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. (Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.)
If $\sigma(M) = 2M$...
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Show that, there do not exist two consecutive perfect numbers.
Show that, there do not exist two consecutive perfect numbers
I know that it is unknown whether there exist any odd perfect number(s) or not.
Also, I know that all the even perfect numbers are ...
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On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.
A number $N$ is said to be perfect if $\sigma(N)=...
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Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/...
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Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II
This post is an offshoot of this MSE question.
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-\...
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If $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1) n^2$, does it follow that $q n^2$ is perfect?
Let $q$ be an (odd) prime, and let $\gcd(q,n)=1$.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
A number $N$ is said to be perfect if $\sigma(N)=2N$.
...
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On the quantity $I(q^k) + I(n^2)$ considered as functions of $q$ and $k$, when $q^k n^2$ is an odd perfect number with special prime $q$ - Part II
Hereinafter, let $N = q^k n^2$ be an odd perfect number with special/Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the abundancy index of the positive integer $x$ ...
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On the conjectured inequality $q > k$, where $q^k n^2$ is an odd perfect number with special prime $q$
Let $N=q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Descartes, Frenicle, and subsequently Sorli conjectured that $k=1$ ...
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On odd perfect numbers and a GCD - Part VI
(Note: This post is closely related to this earlier MSE question.)
Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$....
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On odd perfect numbers and a GCD - Part V
(Note: This post is tangentially related to this earlier MSE question.)
Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,...
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