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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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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 $M$ ...
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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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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 ...
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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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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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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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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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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 ...
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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 ...
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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 ...
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 ...