Puzzle: The set with the bad element removed.

The following is a puzzle that I have posed to various usenet groups over the years. As yet I have recieved no correct answers, so I am posing it here.

The puzzle is in a standard form: First determine the defining characteristic of a set of numbers and, second, find a missing member of the set.

There's no prize, reward, etc for solving the puzzle. It is offered only for the enjoyment of those who take pleasure from wrangling with this sort of thing.

Enjoy.


    The Set H:
  1. Consider the following group of integers:

    ( 10 173 190 2766 2781 3053 3054 3243 3245 3499 3501 3771 4013 4077 4078 43966 44013 47806 47838 48813 48879 51966 52958 57005 57007 57069 64206 64222 65261 712173 11325150 14600926 14613198 15727310 16435934 )

    This list contains all but one of the known elements of a set (see below).
    Let us call this set H.

  2. Membership in the set H is defined by a single, simple defining characteristic which all of its elements share.

    The element X:

  3. One member of the set H has been excluded from the list above. Call this element X.

  4. Furthmore, while X is certainly a fully legal element of H, it could be described as being the "bad" member of the set, in a way that becomes obvious once the defining chararcteristic of the set H is understood.

  5. Finally, it can be shown that X is the only element of H with the property of being "bad".

    Other facts:

  6. The smallest element of the set H is the number 10 .
    Note: This assertion can be proved.

  7. Moreover, with the exception of the single missing element, I believe the above list to contain all the elements of the set H.
    Note: I have no proof for this assertion. Indeed other elements may exist. In any case the truth of this assertion does not affect the puzzle.

The Puzzle:
After determining the defining characteristic of the set H, find this missing element X.


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