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On OPNs and SOPNs

(I hope that this question is appropriate for this site. If it is not, please feel free to point it out and I will then cross-post to MSE.) OPNs are odd perfect numbers. SOPNs are spoof odd perfect ...
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1answer
137 views

Does there exist an integer that is both solitary and almost perfect?

This question is an offshoot from the following MSE post. I hope that it is appropriate for this site. Let $\sigma(x)$ be the sum of the divisors of $x$. An integer $a$ is said to be solitary if ...
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1answer
322 views

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

(Note: This was cross-posted from MSE.) 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 $\gcd(q, n) = 1$). Therefore, ...
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1answer
238 views

A Problem Concerning Odd Perfect Number

Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number. I know there are results much stronger than the one above, but I am looking for an answer ...
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1answer
112 views

Solutions of $rad(\sigma(m))=2rad(m)$

For $m$ a positive integer greater than $1$, let $rad(m)$ be the product of all distinct primes dividing $m$. If $n$ is an odd perfect number (conjectured not to exist), one would have $\sigma(n)=2n$, ...
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1answer
187 views

An equation involving perfect numbers

Let $s,x_1,x_2,\cdots, x_s$ be natural numbers not neccesarily distinct. I am interested in solving the equation $$(x_1+x_2+\cdots +x_s)^s=2^s(x_1\cdot x_2\cdots x_s)^2$$ Some Notes: I have found ...
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1answer
255 views

Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?

I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...
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1answer
275 views

Any results towards the irrationality of the sum of reciprocals of perfect numbers? [closed]

This question is a follow up to my comment to Sum of the reciprocal of perfect numbers. I would like to know which results have been published about the possible irrationality of the sum of ...
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134 views

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$. $\sigma_k(n)$ is the division function and $\sigma(n)=\sigma_1(n)$. A number is ...
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1answer
172 views

Are all known $k$-multiperfect numbers (for $k > 2$) not squarefree?

I asked the following question in MSE four ($4$) days ago, but so far nobody has posted an answer. The gist of the question is as follows: Are all known $k$-multiperfect numbers (for $k > ...
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1answer
412 views

What is the latest progress in the research on Odd Perfect numbers? [closed]

What is the latest progress in the research on Odd Perfect numbers? I may be wrong, but I found a little on Perfect numbers in the latest issues of SCI journals. Is it really so? I would like to have ...
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182 views

Proof of the Infinitude of Odd Primitive Pseudoperfect Numbers

I'm interested in the infinitude of odd primitive pseudoperfect numbers. Richard K. Guy's book "Unsolved Problems in Number Theory 3rd edition" says that P. Erdős proved the infinitude of odd ...
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3answers
280 views

Integers n such that sigma(n)=omega(n)n and omega(n) divides n

Are there other integers $n$ than even perfect numbers such that $\sigma(n)=\omega(n)n$ and $\omega(n)\vert n$? Thanks in advance.