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:

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.

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

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

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.

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

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

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.
6FEB97
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