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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 $\sigma(n)=2n$, hence $rad(\sigma(n))=2rad(n)$. Has this equation been considered so far? Are there any known solutions to it?
Thanks in advance.

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  • $\begingroup$ The equation is close to defining n as an odd k-multiperfect where k rational happens to be composed of prime factors n. An even weaker condition is P(rad(sigma(n))) <= P(n), where P is the largest prime factor of n > 1. As far as I know, even this weaker condition for n a prime power has not been investigated. Gerhard "Would Like To See Answers" Paseman, 2014.02.27 $\endgroup$ Feb 27 '14 at 17:58
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    $\begingroup$ It sounds like you will find the (very similar) problems considered in this paper of interest: math.uga.edu/~pollack/pperfs16.pdf $\endgroup$ Feb 27 '14 at 18:36
  • $\begingroup$ This MSE question might be related. $\endgroup$ Nov 12 '15 at 20:34
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The smallest solution to your equation ${\rm rad}(\sigma(n)) = 2{\rm rad}(n)$ is $n = 135$.

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  • $\begingroup$ Thanks! Can it be proven that there are only finitely many solutions? $\endgroup$ Feb 27 '14 at 18:11
  • $\begingroup$ @SylvainJULIEN: Why would you expect this to be true? -- By the way, the further solutions $n < 1000000$ are 891, 200655, 307125, 544635. $\endgroup$
    – Stefan Kohl
    Feb 27 '14 at 18:16
  • $\begingroup$ I don't really expect it to be true. It would just be interesting to know whether there are finitely many solutions or not, so one has to pick up one of both possibilities. $\endgroup$ Feb 27 '14 at 18:27
  • $\begingroup$ And if it ever turns out that there are finitely many solutions, this will entail that there are finitely many OPNs. $\endgroup$ Feb 27 '14 at 18:30
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    $\begingroup$ The sequence has now been entered into the online encyclopedia, oeis.org/A238330, with a link back here. $\endgroup$ Feb 27 '14 at 22:01

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