Yes, it is probably a phraseology problem or a misunderstanding, but I still don't agree with this sentence:

All of them can be defined as the sums of multiples of powers of 3. But only the first number in each block is a power of 3.

To me, these numbers are all sums of single powers of 3. For example, taking the beginning of your list:

1          -- 3**0
3 4        -- 3**1, 3**0 + 3**1
9 10 12 13 -- 3**2, 3**0 + 3**2, 3**1 + 3**2, 3**0 + 3**1 + 3**2
etc.
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So they are all sums of pure (or single) powers of 3, not sums of multiples of powers of 3 (which would imply numbers expressed with other digits than 0 and 1 in base 3). And so is 82000.

And I agree with you that you don't have to consider these other numbers, only those that are pure powers of 3 are of interest for the search; so, as you said, only the first one of each block if you want to figure out whether 82000 or any other number qualifies the test.