Find all positive integers $k$ for which there exist $a$, $b$, and $c$ positive integers such that \[\lvert (a-b)^3+(b-c)^3+(c-a)^3\rvert=3\cdot2^k.\]
1. **Factor the given expression:**
   We start with the given equation:
   \[
   \lvert (a-b)^3 + (b-c)^3 + (c-a)^3 \rvert = 3 \cdot 2^k
   \]
   Notice that the sum of cubes can be factored using the identity:
   \[
   (a-b)^3 + (b-c)^3 + (c-a)^3 = 3(a-b)(b-c)(c-a)
   \]
   Therefore, the equation becomes:
   \[
   \lvert 3(a-b)(b-c)(c-a) \rvert = 3 \cdot 2^k
   \]

2. **Simplify the equation:**
   Divide both sides by 3:
   \[
   \lvert (a-b)(b-c)(c-a) \rvert = 2^k
   \]

3. **Substitute variables:**
   Let \( a-b = x \), \( b-c = y \), and \( c-a = z \). Then we have:
   \[
   x + y + z = 0
   \]
   This implies \( z = -(x + y) \). Substituting \( z \) into the equation, we get:
   \[
   \lvert x y (-(x + y)) \rvert = 2^k
   \]
   Simplifying further:
   \[
   \lvert -x y (x + y) \rvert = 2^k
   \]
   Since the absolute value of a negative number is positive, we have:
   \[
   \lvert x y (x + y) \rvert = 2^k
   \]

4. **Analyze the powers of 2:**
   For \( \lvert x y (x + y) \rvert \) to be a power of 2, \( x \), \( y \), and \( x + y \) must all be powers of 2. Let:
   \[
   x = 2^n \quad \text{and} \quad y = 2^m
   \]
   Then:
   \[
   x + y = 2^n + 2^m
   \]
   For \( 2^n + 2^m \) to be a power of 2, \( n \) must equal \( m \) (since the sum of two distinct powers of 2 is not a power of 2). Therefore:
   \[
   x = y = 2^n
   \]
   Thus:
   \[
   x + y = 2^{n+1}
   \]

5. **Calculate the product:**
   Substitute \( x = y = 2^n \) into the product:
   \[
   \lvert 2^n \cdot 2^n \cdot 2^{n+1} \rvert = 2^{n} \cdot 2^{n} \cdot 2^{n+1} = 2^{3n+1}
   \]
   Therefore:
   \[
   k = 3n + 1
   \]

Conclusion:
The positive integers \( k \) for which there exist positive integers \( a \), \( b \), and \( c \) such that \(\lvert (a-b)^3 + (b-c)^3 + (c-a)^3 \rvert = 3 \cdot 2^k\) are those of the form \( k = 3n + 1 \) for some non-negative integer \( n \).

The final answer is \( \boxed{ k = 3n + 1 } \) for some non-negative integer \( n \).