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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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5
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1answer
110 views

Is the odd part of even almost perfect numbers (other than the powers of two) not almost perfect?

Let $\sigma(x)$ denote the sum of the divisors of $x$. A number $M$ is called almost perfect if $\sigma(M) = 2M - 1$. If $M$ is an even almost perfect number, then the only known examples for $M$ ...
2
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1answer
169 views

Do there exist two perfect numbers $N_1$ and $N_2$ such that $N_1 + N_2$ is also perfect?

Let $\sigma(x)$ be the sum of the divisors of $x$. So, for example, $\sigma(6) = 1 + 2 + 3 + 6 = 12$. A positive integer $N$ is said to be perfect if $\sigma(N) = 2N$. Consider the equation $$\...
0
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1answer
130 views

Prove that an odd perfect number has 3 distinct primes

So far i've tried looking at an equation which is true for perfect numbers: $\sigma(n) = 2n$. where $\sigma(n) = \sum_{d|n} d$ i started looking at a odd perfect number which has 1 prime divisor say ...
3
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2answers
461 views

Odd perfect numbers must have an odd number of proper divisors

Theorem: If an odd perfect number exists, then it has an odd number of odd proper divisors. $(1)$ Assume that an odd perfect number exists. Call it $n$. $(2)$ $2$ is not a divisor because $n$ is odd....
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2answers
179 views

Perfect numbers

Define a Perfect (capital-P) number as a natural number that is equal to the sum of its Divisors excluding 1 and the number itself. (So the Divisors of 28 are 2, 4, 7, 14, summing to 27.) Is there any ...
4
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1answer
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 ...
4
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1answer
170 views

Perfect number in gaussian integers

We have complete description about irreducibles in the ring Z[i],of gaussian integers. Now I was trying to define suitably the notion of "perfect number" in Z[i]. But the problem is unique ...
2
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2answers
163 views

When does :$\sigma(\sigma(2n))=\sigma(\sigma(n))$ and $\sigma(n)$ is sum divisors of the positive integer $n$?

Is there someone who can show me When does: $$\sigma(\sigma(2n))=\sigma(\sigma(n))$$ where : $\sigma(n)$ denotes the sum of divisors of the positive integer $n$ ? Note (1) : I came across this ...
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2answers
74 views

How do I show $1$ is not a trivial odd perfect number?

This question related to my this question in MO ,some comments stated that the integer $1$ is trivial answer for this question ,but here i'm very confused when we say that the sum divisors of $1$ is $...
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2answers
136 views

I attempt integrate another factor 2 in the definition of even perfect numbers

I use the method display by Florian in [1] (in true both statments of this problem are due to Florian at 99%) to compute from $\sigma(2n)-(\sigma(n)+\sigma(n))=2^p$ (where $\sigma$ is the sum of ...
2
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4answers
139 views

the product of an odd perfect number and some even perfect number is perfect

If $a$ were an odd perfect number ,does there exist an even perfect number $b$ such that $ab$ is a perfect number?
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1answer
91 views

Is $p(n) = 2^{n²+n-1} - n² - n + 1$ abundant for all $n >1$?

Let $p(n)=2^{n²+n-1}-n²-n+1 $, and let $\delta(n)$ be sum of proper divisors of $n\in\mathbb{N}$. After some verifications according to the values of $n>1$ I noticed: $$\delta(p(n))> p(n)$$ ...
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1answer
180 views

Prove that there exist $2015$ consecutive abundant numbers [closed]

A positive integer $N$ is called abundant if the sum of its divisors is greater than $N$: $\delta (N) >N$. My question is: Prove that there exists an integer: $k\in\mathbb N\setminus\{0\}$ ...
2
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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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2answers
131 views

Proof that odd perfect numbers cannot consist of single unique factors?

I'm a high school student, so please point out my mistakes nicely :) So we already know odd perfect numbers cannot be in the form of a square, but how about that they cannot be in this form: $$P=abcd....
2
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1answer
89 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

(Note: This question has been cross-posted from MO.) Let $\sigma$ be the classical sum-of-divisors function. A number is said to be perfect if $\sigma(N)=2N$. If $q^k n^2$ is an odd perfect number ...
5
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1answer
177 views

If $q^k n^2$ is an odd perfect number with Euler factor $q^k$, can $q = 73$ hold?

Call a number $N$ perfect if $\sigma(N)=2N$ where $\sigma$ is the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number, can $q = 73$ hold? Here is my attempt: Since $37=(...
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1answer
334 views

Prove, if $n>6$ is an even, perfect number, then $n\equiv4 \pmod 6$

I've been working on this for quite awhile, and am stumped after a little bit. I have some stuff written down, but I just don't know how to completely prove it. I don't have much done yet: $(2^{p-1})(...
2
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1answer
103 views

Is it conjectured that there are no odd multi-perfect numbers?

It is conjectured that there is no odd perfect number. But is there a stronger conjecture that there are no odd multi-perfect numbers ? Wikipedia shows a useful link, but my conjecture is not ...
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2answers
214 views

Every superperfect number(except $2$) is a square number?

I recently read about superperfect numbers: $σ^2(n) = 2n$, where $σ(n)$ is the divisor function. I saw that the first few numbers were: $2, 4, 16, 64, 4096, 65536, 262144$, which are all square-...
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1answer
224 views

What is known about multi-perfect numbers?

It is unknown if odd perfect numbers exist and it is known that the even perfect numbers are those of the form $$2^{n-1}(2^n-1)$$, where $2^n-1$ is a (Mersenne-)prime. But what is known about the ...
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1answer
75 views

Is every multiple of a perfect number a semiperfect number? If not, when isn't it?

Is every multiple of a perfect number a semiperfect number? If not, when isn't it? A perfect number is a number that is equal to the sum of its proper divisors (divisors not including the number ...
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0answers
77 views

Largest known multi-perfect number (excluding perfect numbers)

What is the largest known multi-perfect number (excluding the perfect numbers) ? [2, 94; 3, 32; 5, 9; 7, 11; 11, 2; 13, 8; 17, 1; 19, 5; 23, 1; 29, 2; 31, 2; 37, 1; 43, 1; 53, 1; 59, 1; 61, 2; 67, 1;...
3
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1answer
119 views

Number $N>6$, such that $N-1$ and $N+1$ are primes and $N$ divides the sum of its divisors

The perfect number $6$ is in the middle of the primes $5$ and $7$. It is the only perfect number with this property because odd numbers are not in the middle of two twin primes and even perfect ...
2
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5answers
227 views

Prove the existence or the non-existence of a couple of numbers ($n$,$m$) such that $n^2=m!$ [duplicate]

In recent days, while I was doing exercises on combinatorics, I thought if a number $m!$ could be a perfect square. I proved to demonstrate it through the prime factorization. My attempt: $$k=m!=p_1^{...
2
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2answers
97 views

How Deficient a Number is? (Finding numbers having a certain deficiency)

This question was edited, in particular equations were corrected: A number N is said to be deficient by an integer $d$ if: $\sigma(N)=2N-d$ Note that powers of 2 are deficient by 1. While a prime $...
2
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1answer
140 views

A question on (odd) perfect numbers

(Note: This has been cross-posted to MO.) Let $\sigma(x)$ be the (classical) sum of the divisors of $x$. A number $N \in \mathbb{N}$ is called perfect if $\sigma(N)=2N$. An even perfect number $U$ ...
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0answers
67 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

(Note: This has been cross-posted to MO.) A positive integer $N$ is said to be perfect if $\sigma(N) = 2N$, where $\sigma(x)$ is the sum of the divisors of $x$. An odd perfect number $N$ is said to ...
0
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0answers
96 views

On odd perfects and spoofs

This question is an offshoot of this MSE post. Let $\sigma$ be the classical sum-of-divisors function. An odd perfect number $N$ is said to be given in Eulerian form if $\sigma(N)=2N$ and $N={q^k}{n^...
4
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1answer
555 views

Relationship between Mersenne Primes and Triangular / Perfect Numbers

I'm a new user and have only a college sophomore's understanding of mathematics, so please bear with me. I was reading a book titled “The Simpsons and their Mathematical Secrets” in which the author ...
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1answer
67 views

Is the Euler prime of an odd perfect number a palindrome (in base $10$), 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 ...
1
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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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0answers
95 views

A question related to the sum-of-divisors function

In what follows, let $\sigma$ be the sum-of-divisors function, and assume that we have $\sigma(a^b)\sigma(c^2)=2{a^b}{c^2}$ together with $\gcd(a,c)=1$ and integers $a, c > 1$. Let $I(x)=\sigma(x)/...
2
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1answer
323 views

Mersenne primes and superperfect numbers

Definition: Let $n\in\mathbb{Z}$ with $n>0$. Then $n$ is said to be superperfect if $\sigma(\sigma(n)) = 2n$. Where $\sigma$ is the sum of positive divisors arithmetic function. ($\sigma(n) = \sum_{...
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4answers
93 views

Databases for perfect numbers

So I have been trying to find a database that offers perfect numbers. I need this to help me and a friend with a project that we have been working on for a while involving the odd perfect number ...
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0answers
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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1answer
388 views

Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.

Define $\sigma(m) = \sum$ d : d|n. Prove that $p^j$$q^i$ cannot be a perfect number for $p, q$ odd, distinct primes. Attempt at Solution: I have shown that $p^k$ can never be a perfect number, and ...
2
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2answers
848 views

Product Perfect Numbers (Number Theory)

I've just figured out that a product perfect number is a number N such that either N=pq or $N=q^3$. But how do I prove this? Attempt: I know that any number of this form is a product perfect number ...
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0answers
50 views

<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. ...
4
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2answers
768 views

Prove that any power of a prime is not a perfect number [closed]

How do I prove: Let $p$ be a prime, and $n$ be a positive integer. Then $p^n$ is not a perfect number. One example is when $p = 2$ and $n = 3$, the question is to show $8$ is not a perfect number....
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2answers
2k views

Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfectnumber (well it ...
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1answer
50 views

On a certain “obvious” implication concerning odd perfect numbers

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$). ($\sigma(x)$ gives the sum of the divisors of $x$, ...
2
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1answer
118 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This has been cross-posted to MO.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). ...
1
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0answers
66 views

Show that the product of the divisors of an even perfect number is $n^p$.

Let $n=(2^{p-1})\cdot(2^p-1)$ where n is an even prefect number and p is prime. I know the divisors are $1, 2, 2^2, 2^3, ..., 2^{p-1}, 2^p-1$, and $n$. I get $(2^{(1/2)p(p-1)}\cdot ((2^p)-1)\cdot(2^{...
1
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1answer
563 views

Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number

Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number. I am little confused about this problem, any insight?
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2answers
321 views

Why does not the perfect number formula imply there are infinitely many perfect numbers?

We know the even perfect number formula is $2^{p-1}(2^p − 1)$ and it is known that the multiplication of a even number and odd number is a even number. So why can't we say there are infinitely many ...
5
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2answers
914 views

Is this proof that there are no perfect, odd, integer square numbers legitimate?

Assumptions: Any even number times any other number is always an even number. An odd number times an odd number is always an odd number. An even number plus an even number is even, and an odd number ...
2
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0answers
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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0answers
99 views

How I could transform this into product over primes :$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+…\frac{1}{2^p-1}$?

1)Can I transforme this sum into product OVER primes:$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+....\frac{1}{2^p-1}$ ? Note : p is prime number and ${2^p-1}$ is prime 2)I would be interest to know ...
2
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0answers
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 ...