cQuents, 9 bytes
&#N_=N+.0
& is broken on TIO right now, so TIO link will not work, but it would time out anyway, as the only way to start at 2^53 is modifing the interpreter.
Explanation
Implicit input n
& Output first n terms of the sequence
# Only include an integer N in the sequence if:
N N
_= is not equal to
N+.0 N converted to float
Stephen
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