Find all positive integers $n$ for which there exist positive integers $a, b,$ and $c$ that satisfy the following three conditions:
$\bullet$ $a+b+c=n$
$\bullet$ $a$ is a divisor of $b$ and $b$ is a divisor of $c$
$\bullet$ $a < b < c$
1. We start with the given conditions:
   - \(a + b + c = n\)
   - \(a\) is a divisor of \(b\) and \(b\) is a divisor of \(c\)
   - \(a < b < c\)

2. Let \(b = p \cdot a\) and \(c = q \cdot b = pq \cdot a\) where \(p\) and \(q\) are integers greater than 1 (since \(a < b < c\)).

3. Substituting \(b\) and \(c\) into the sum \(a + b + c = n\), we get:
   \[
   a + p \cdot a + pq \cdot a = n
   \]
   \[
   a(1 + p + pq) = n
   \]

4. Since \(a\) is a positive integer, \(1 + p + pq\) must be a divisor of \(n\). We need to find the minimum value of \(1 + p + pq\):
   - For \(p = 2\) and \(q = 2\), \(1 + 2 + 2 \cdot 2 = 7\)
   - For \(p = 2\) and \(q = 3\), \(1 + 2 + 2 \cdot 3 = 9\)
   - For \(p = 3\) and \(q = 2\), \(1 + 3 + 3 \cdot 2 = 10\)
   - For \(p = 3\) and \(q = 3\), \(1 + 3 + 3 \cdot 3 = 13\)

5. Therefore, the smallest possible value of \(1 + p + pq\) is 7. This implies that \(n\) must be at least 7.

6. We now need to check if \(n = 7\) is achievable:
   - Let \(a = 1\), \(b = 2\), and \(c = 4\). Then \(a + b + c = 1 + 2 + 4 = 7\), which satisfies all conditions.

7. For \(n = 8\):
   - We need \(a(1 + p + pq) = 8\). The possible values for \(1 + p + pq\) are 7, 8, 9, 10, etc.
   - Since 8 is not achievable with \(1 + p + pq\) being 7, 9, or 10, \(n = 8\) is not possible.

8. For \(n = 12\):
   - We need \(a(1 + p + pq) = 12\). The possible values for \(1 + p + pq\) are 7, 9, 10, 12, etc.
   - Since 12 is not achievable with \(1 + p + pq\) being 7, 9, or 10, \(n = 12\) is not possible.

9. For \(n = 24\):
   - We need \(a(1 + p + pq) = 24\). The possible values for \(1 + p + pq\) are 7, 9, 10, 12, 13, etc.
   - Since 24 is not achievable with \(1 + p + pq\) being 7, 9, 10, 12, or 13, \(n = 24\) is not possible.

10. For other values of \(n\), we can find suitable \(a, b, c\) such that \(a + b + c = n\) and \(a\) divides \(b\) and \(b\) divides \(c\).

Conclusion:
All positive integers \(n\) for which there exist positive integers \(a, b, c\) that satisfy the given conditions are:
\[
n \in \mathbb{N} \setminus \{1, 2, 3, 4, 5, 6, 8, 12, 24\}
\]

The final answer is \( \boxed{ \mathbb{N} \setminus \{1, 2, 3, 4, 5, 6, 8, 12, 24\} } \).