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This puzzle is based off the What is a Word™, What is a Phrase™ and What is a Number™ series started by JLee.

If a number conforms to a certain rule, I call it a Self-Rejecting Number™. Use the following examples to find the rule:

Lists of Self-Rejecting Numbers™ and Non-Self-Rejecting Numbers™

Here is a CSV:

Self-Rejecting Numbers™,Non-Self-Rejecting Numbers™
836,638
7192,9172
7912,1972
13930,13931
17272,27171
570988,988570
999670,666970
1811110,1111810
3123470,2432170
3432170,2123470
5555690,5555960
6035084,4805306

The puzzle relies on the series' inbuilt assumption, that each number can be tested for whether it is a Self-Rejecting Number™ without relying on the other numbers.

These are not the only examples of Self-Rejecting Numbers™, more can be found.

Hint 1 (small):

It is an open question in mathematics whether any are odd.

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  • $\begingroup$ This is my first puzzle on Puzzling SE, so please let me know whether it's a good one! :) $\endgroup$ – boboquack Oct 27 '16 at 6:41
  • $\begingroup$ Any reason why the left list is in order but not the right list? $\endgroup$ – stack reader Oct 27 '16 at 6:45
  • $\begingroup$ @stackreader It's just so the numbers on the left and right bear a resemblance to each other. As in the statement, each number can be tested for whether it is a Self-Rejecting Number™ without relying on any other numbers. $\endgroup$ – boboquack Oct 27 '16 at 6:51
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    $\begingroup$ Are there examples of Self-Rejecting Numbers that are odd? $\endgroup$ – Joe Oct 27 '16 at 7:06
  • $\begingroup$ @Joe see my hint - if you want one $\endgroup$ – boboquack Oct 27 '16 at 7:43
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Self-Rejecting Numbers™ are ...

weird.
no. really. they're weird.

And they are Self-Rejecting™ because ...

their definition is such that the sum of their divisors is greater than the number itself, but no subset of those divisors adds exactly to the number itself; that is, no set of their divisors can add up to themselves, so in this way they never match themselves (and thus self-reject).

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  • $\begingroup$ I have three questions: How did you get to this answer? Did you think this was a good question? Can you explain the name Self-Rejecting? $\endgroup$ – boboquack Oct 27 '16 at 8:19
  • $\begingroup$ The "small" hint actually is what led me to the solution. Well, that and this handy list of unsolved problems in mathematics and a quick search on that page for the word "odd". Only two things turned up - are there any odd perfect numbers, and are there any odd weird numbers. Looking at both, it was trivial to find that your list came from the list of weird numbers. $\endgroup$ – Rubio♦ Oct 27 '16 at 8:22
  • $\begingroup$ Should I make more of this type of question? (I have only seen 2 posted, and 1 was deleted) $\endgroup$ – boboquack Oct 27 '16 at 8:25
  • $\begingroup$ Correct in why they are self-rejecting $\endgroup$ – boboquack Oct 27 '16 at 8:28
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    $\begingroup$ In all honesty I think this question devolves to math trivia, and without the hint it probably wouldn't be solvable. Questions like that generally don't make good puzzles since they can't really be reasoned out or guessed at in anyone's lifetime. A puzzle should be solvable without knowing its solution. $\endgroup$ – Rubio♦ Oct 27 '16 at 8:29

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