Find all integer positive numbers $n$ such that:
$n=[a,b]+[b,c]+[c,a]$, where $a,b,c$ are integer positive numbers and $[p,q]$ represents the least common multiple of numbers $p,q$.
1. **Lemma:**
   We need to show that \( n \) can be written in the form \( n = [a,b] + [b,c] + [c,a] \) if and only if \( 2n \) can be written in that form.

   **Proof:**
   - If \( n = [a,b] + [b,c] + [c,a] \), then:
     \[
     2n = 2([a,b] + [b,c] + [c,a]) = [2a,2b] + [2b,2c] + [2c,2a]
     \]
     This follows because the least common multiple (LCM) of \( 2a \) and \( 2b \) is \( 2 \times \text{LCM}(a,b) \), and similarly for the other terms.

   - Conversely, if \( 2n = [a,b] + [b,c] + [c,a] \), we consider two cases:
     - If \( 2 \mid a, b, c \), then let \( a = 2a', b = 2b', c = 2c' \). We have:
       \[
       2n = [2a', 2b'] + [2b', 2c'] + [2c', 2a'] = 2([a', b'] + [b', c'] + [c', a'])
       \]
       Thus, \( n = [a', b'] + [b', c'] + [c', a'] \).

     - If \( 2 \mid a, b \) but \( 2 \nmid c \), then let \( a = 2a', b = 2b' \). We have:
       \[
       2n = [2a', 2b'] + [2b', c] + [c, 2a'] = 2([a', b'] + [b', c] + [c, a'])
       \]
       Thus, \( n = [a', b'] + [b', c] + [c, a'] \).

   Therefore, the lemma is proven. \(\blacksquare\)

2. Now, consider \( n = 2m + 1 \). We need to show that \( n \) cannot be written in the form \( n = [a,b] + [b,c] + [c,a] \).

   - Suppose \( n = 2m + 1 \). Then:
     \[
     n = [m,1] + [1,1] + [1,m]
     \]
     However, the LCM of any two positive integers is at least as large as the larger of the two integers. Therefore:
     \[
     [m,1] + [1,1] + [1,m] \geq m + 1 + 1 + m = 2m + 2 > 2m + 1
     \]
     This implies that \( n = 2m + 1 \) cannot be written in the form \( n = [a,b] + [b,c] + [c,a] \).

3. Using the lemma, we conclude that \( n \) can be written in the form \( n = [a,b] + [b,c] + [c,a] \) if and only if \( n \) is not a power of 2.

The final answer is \( \boxed{ n } \) can be written in the form \( n = [a,b] + [b,c] + [c,a] \) if and only if \( n \) is not a power of 2.