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Can an odd perfect number be divisible by either $2049$ or $2051$?

Note that $2049 = 3 \cdot {683}$, and that $2051 = 7 \cdot {293}$.

Added July 15 2016

It is known that an odd perfect number cannot be divisible by 105, 825, and 5313. It is also tempting to conjecture that an odd perfect number cannot be divisible by 165, but this has not yet been completely proved.

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Jose Arnaldo Bebita Dris
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    How would this Question be answered when the existence of any odd perfect number is an open problem? Potentially one or both of these possibilities can be ruled out, because there are a number of results on prime divisors of any odd perfect number, but my impression is that the results of this kind do not exclude odd prime factors (but rather require several prime divisors). – hardmath Jul 11 '16 at 11:30
  • See Wolfram MathWorld: Odd Perfect Number for several of these results about the number of prime divisors and the size of the smallest prime divisor for any possible odd perfect number. – hardmath Jul 11 '16 at 11:38
  • @hardmath, I invite you to check out http://mathoverflow.net/questions/178286, if you haven't done so already. =) – Jose Arnaldo Bebita Dris Jul 11 '16 at 12:13
  • Additionally, it is known that an odd perfect number cannot be divisible by 105, 825, and 5313. It is also tempting to conjecture that an odd perfect number cannot be divisible by 165, but this has not yet been completely proved. – Jose Arnaldo Bebita Dris Jul 13 '16 at 14:53
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    In my opinion adding these links to the body of your Question would improve it. The observation that no odd perfect number can be divisible by $105$ may serve Readers as an illustration of the type of proof you are looking for (showing abundance exceeds $2$). – hardmath Jul 14 '16 at 12:53

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