The [b]Collaptz function[/b] is defined as $$C(n) = \begin{cases} 3n - 1 & n\textrm{~odd}, \\ \frac{n}{2} & n\textrm{~even}.\end{cases}$$ 
We obtain the [b]Collaptz sequence[/b] of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of $13$ begins with $13, 38, 19, 56, 28, \cdots$ and so on. Find the sum of the three smallest positive integers $n$ whose Collaptz sequences do not contain $1,$ or in other words, do not [b]collaptzse[/b].

[i]Proposed by Andrew Wu and Jason Wang[/i]
To solve the problem, we need to find the sum of the three smallest positive integers whose Collaptz sequences do not contain the number 1. We will use the definition of the Collaptz function and check each integer sequentially to determine if its sequence contains the number 1.

1. **Check \( n = 1 \):**
   \[
   C(1) = 3 \cdot 1 - 1 = 2
   \]
   The sequence is \( 1, 2 \), which contains 1. Therefore, 1 collaptzses.

2. **Check \( n = 2 \):**
   \[
   C(2) = \frac{2}{2} = 1
   \]
   The sequence is \( 2, 1 \), which contains 1. Therefore, 2 collaptzses.

3. **Check \( n = 3 \):**
   \[
   C(3) = 3 \cdot 3 - 1 = 8 \\
   C(8) = \frac{8}{2} = 4 \\
   C(4) = \frac{4}{2} = 2 \\
   C(2) = \frac{2}{2} = 1
   \]
   The sequence is \( 3, 8, 4, 2, 1 \), which contains 1. Therefore, 3 collaptzses.

4. **Check \( n = 4 \):**
   \[
   C(4) = \frac{4}{2} = 2 \\
   C(2) = \frac{2}{2} = 1
   \]
   The sequence is \( 4, 2, 1 \), which contains 1. Therefore, 4 collaptzses.

5. **Check \( n = 5 \):**
   \[
   C(5) = 3 \cdot 5 - 1 = 14 \\
   C(14) = \frac{14}{2} = 7 \\
   C(7) = 3 \cdot 7 - 1 = 20 \\
   C(20) = \frac{20}{2} = 10 \\
   C(10) = \frac{10}{2} = 5
   \]
   The sequence is \( 5, 14, 7, 20, 10, 5 \), which repeats without containing 1. Therefore, 5 does not collaptzse.

6. **Check \( n = 6 \):**
   \[
   C(6) = \frac{6}{2} = 3
   \]
   Since 3 collaptzses, 6 also collaptzses.

7. **Check \( n = 7 \):**
   \[
   C(7) = 3 \cdot 7 - 1 = 20 \\
   C(20) = \frac{20}{2} = 10 \\
   C(10) = \frac{10}{2} = 5
   \]
   Since 5 does not collaptzse, 7 also does not collaptzse.

8. **Check \( n = 8 \):**
   \[
   C(8) = \frac{8}{2} = 4 \\
   C(4) = \frac{4}{2} = 2 \\
   C(2) = \frac{2}{2} = 1
   \]
   The sequence is \( 8, 4, 2, 1 \), which contains 1. Therefore, 8 collaptzses.

9. **Check \( n = 9 \):**
   \[
   C(9) = 3 \cdot 9 - 1 = 26 \\
   C(26) = \frac{26}{2} = 13 \\
   C(13) = 3 \cdot 13 - 1 = 38 \\
   C(38) = \frac{38}{2} = 19 \\
   C(19) = 3 \cdot 19 - 1 = 56 \\
   C(56) = \frac{56}{2} = 28 \\
   C(28) = \frac{28}{2} = 14 \\
   C(14) = \frac{14}{2} = 7
   \]
   Since 7 does not collaptzse, 9 also does not collaptzse.

Thus, the three smallest positive integers that do not collaptzse are 5, 7, and 9. Their sum is:
\[
5 + 7 + 9 = 21
\]

The final answer is \(\boxed{21}\)