Determine the smallest positive integer $n$ whose prime factors are all greater than $18$, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.
1. We start with the equation \( n = a^3 + b^3 \). Using the identity for the sum of cubes, we have:
   \[
   n = a^3 + b^3 = (a + b)(a^2 - ab + b^2)
   \]
   Since all prime factors of \( n \) must be greater than 18, \( a + b \) must be at least 19.

2. To minimize \( n \), we should minimize \( a^3 + b^3 \) while ensuring \( a + b \geq 19 \). We first check the smallest possible value for \( a + b \), which is 19.

3. We test pairs \((a, b)\) such that \( a + b = 19 \). The goal is to find the pair that minimizes \( a^3 + b^3 \).

4. First, we check \((a, b) = (9, 10)\):
   \[
   9^3 + 10^3 = 729 + 1000 = 1729
   \]
   Factoring 1729:
   \[
   1729 = 7 \times 13 \times 19
   \]
   Since 7 and 13 are less than 18, this does not satisfy the condition.

5. Next, we check \((a, b) = (8, 11)\):
   \[
   8^3 + 11^3 = 512 + 1331 = 1843
   \]
   Factoring 1843:
   \[
   1843 = 19 \times 97
   \]
   Both 19 and 97 are greater than 18, so this satisfies the condition.

6. We conclude that \( n = 1843 \) is the smallest possible \( n \) such that all its prime factors are greater than 18 and it can be expressed as \( n = a^3 + b^3 \) with positive integers \( a \) and \( b \).

The final answer is \( \boxed{1843} \).