2020 Mathcounts State Sprint Round #30

Solution: It is important to note that $729=3^6$. If we take a look at the smaller powers of 3, namely $3^i, i = 0,1,2,3$, we might be able to find a pattern. With $3^0$, it is obvious: 1 is the only number that exists in the sequence, and for $3^1$ there are two: 1, 2. For $3^2$, the problem lists them for us: 1,2,4,5, which is 4. We can notice a pattern here: For $3^0$ it is 1, for $3^1$ it is 2, for $3^2$ it is 4, so let’s check for $3^3$. Continuing the sequence, we get
$$1,2,4,5,10,11,13,14$$
There are 8 terms here, so we definitely have a pattern. For each power of 3, the number of sequence elements is two to the power of the same number. So for the first 729 terms, the number of terms is $2^6=\boxed{64}$.