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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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### Relations between composite numbers, prime numbers, and perfect numbers.

(1) A composite number a is a positive integer number that is greater than 1 and can be expressed as the product of two smaller positive integer numbers, say b and c. This definition restricts b and c ...
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### Number in binary as a product

I agree that any binary number that consists of $n$ ones (and no zeros) has as its decimal equivalent the number $2^n - 1$. However, the author of the book I'm reading next makes the following claim, ...
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### If $q^k n^2$ is an odd perfect number with special prime $q$, is it true that $\sigma(n^2)/q^k \mid n^2$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Question If $q^k n^2$ is an odd perfect number with special prime $q$, ...
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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 ...
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### What makes it so difficult to prove that $3$ does not divide an odd perfect number? [closed]

Let $\sigma(x)$ denote the sum of divisors the positive integer $x$. The abundancy index of $x$ is then given by the formula $I(x)=\sigma(x)/x$. A number $N \in \mathbb{N}$ is said to be perfect ...
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### Question on Potential Relationship between Mertens Function and Perfect Number Counting Function

This question assumes the following definitions where $M(x)$ is the Mertens function and $f(x)$ is the perfect number counting function. See here for more information on the Wolfram Language ...
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### What is wrong with my proof of there is no odd perfect number?

The Conjecture: $A \equiv_2 1$ Implies $\sigma(A) \not = 2A$ The Proof: Let $H :=$ $[A = 2Q + 1]$ $\land$ $[\sigma(A) = 2A]$. So $H$ is true IFF $$\sigma(2Q + 1) = 2(2Q + 1).$$ Therefore we will ...
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### Does $2r - \sigma(r)$ divide $\sigma(r)$ if $r$ is deficient-perfect?

The present question is tangentially related to this earlier one. My question here is: Does $2r - \sigma(r)$ divide $\sigma(r)$ if $r$ is deficient-perfect? Recall that a positive integer $x$ is ...
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### Can these bounds in terms of the abundancy index and deficiency functions be improved for deficient-perfect numbers?

Let $$\sigma(x) = \sum_{e \mid x}{e}$$ denote the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$, and the deficiency of $x$ by $D(x)=2x-\sigma(x)$...