4
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Given a positive integer $n$, let $\sigma(n)$ denote the sum of divisors of $n$. We say that a positive integer $n$ is a near-perfect number if $\sigma(n) - 2n$ is greater than $0$ and a proper divisor of $n$.

Question: Is it true that there are only finitely many near-perfect numbers with more than $4$ distinct prime divisors?

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    $\begingroup$ Did you mean "finite number of even near-perfect numbers"? Also, your special number has 4 different prime divisors (not more than 4) -- how it's relevant? Also, see oeis.org/A181595 $\endgroup$ – Max Alekseyev Apr 9 '16 at 5:07
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    $\begingroup$ i say more than 4 prime factors $\endgroup$ – qian feng Apr 9 '16 at 6:09
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    $\begingroup$ I don't see any argument against the value of the q.f.'s question. Thus, I'd like to keep it open. $\endgroup$ – Włodzimierz Holsztyński Apr 9 '16 at 9:17
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    $\begingroup$ Having in mind there are 2 solutions in the first 200 near perfect of this kind, I doubt this is true. Is it unconditionally known there are infinitely many near perfect of this kind (dropping the restriction about 4 prime factors)? $\endgroup$ – joro Apr 9 '16 at 11:22
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    $\begingroup$ contain just four distinct prime divisors. $\endgroup$ – qian feng Apr 9 '16 at 14:08

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