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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 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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### 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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### 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 $119$.
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 ...
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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### 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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### 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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### 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 ...
### 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 ...
### 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 \$...