Let $\sigma(n)$ be the sum of positive divisors of $n$. If $\sigma(n)=2n$ we have that $n$ is one of the perfect numbers as $6$, $28$, $496$ and in general $2^{p-1}\cdot(2^p-1)$, where $2^p-1$ is prime.
What is known about the case $\sigma(n)=kn$ with $k\neq 2$?
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3
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3$\begingroup$ This question is too broad. At any rate, there has been a lot of research on these numbers, see en.wikipedia.org/wiki/Multiply_perfect_number $\endgroup$– GH from MOCommented Oct 9 at 3:43
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1$\begingroup$ If $k$ is meant to be real, then for $k \lt 2$ please see Wikipedia's Deficient number, while for $k \gt 2$, see Abundant number, and the related Highly abundant number. $\endgroup$– John OmielanCommented Oct 9 at 3:47
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$\begingroup$ A list of such numbers is tabulated here. $\endgroup$– Jose Arnaldo BebitaCommented Nov 30 at 12:50
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