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Sequence includes only multiples of Fermat numbers (sequence A080307), but not all of them. It is not certain that A080309 is infinite, but it seems likely given that exactly one-half of all integers are multiples of Fermat numbers (see A080307).
The first 3 even numbers are 2, 4 and 6; the first 3 multiples of Fermat numbers (the numbers of the form 2^(2^n)+1) are 3, 5 and 6. The third even number is also the third Fermat multiple; thus 3 is in the sequence.