For a finite set $A$ of positive integers and its subset $B$, call $B$ a [i]half subset[/i] of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.
1. **Understanding the Problem:**
   We need to determine all positive integers \( n \) such that there exists a finite set \( A \) which has exactly \( n \) half subsets. A subset \( B \) of \( A \) is called a half subset if it satisfies the equation:
   \[
   \sum_{a \in A} a = 2 \sum_{b \in B} b
   \]
   This implies that the sum of the elements in \( B \) is half the sum of the elements in \( A \).

2. **Initial Observation:**
   If \( B \) is a half subset of \( A \), then \( A \setminus B \) is also a half subset of \( A \). This is because:
   \[
   \sum_{a \in A} a = 2 \sum_{b \in B} b \implies \sum_{a \in A} a = 2 \left( \sum_{a \in A} a - \sum_{b \in B} b \right)
   \]
   Simplifying, we get:
   \[
   \sum_{a \in A} a = 2 \sum_{a \in A \setminus B} a
   \]
   Therefore, the number of half subsets must be even.

3. **Constructing an Example for \( n = 2k \):**
   We need to construct a set \( A \) with exactly \( 2k \) half subsets. Let \( 0 < a_1 < a_2 < \dots < a_k \) be any set of rapidly increasing positive integers, for instance, satisfying:
   \[
   a_1 + a_2 + \dots + a_i < a_{i+1}, \quad i = 1, 2, \dots, k-1
   \]
   Take \( S_1 \) large enough and define:
   \[
   a_{k+1+i} := S_1 - a_{k-i}, \quad i = 0, 1, \dots, k-1
   \]
   Let \( r := (k-2)S_1 \). Finally, define the set \( A \) as:
   \[
   A := \{r, a_1, a_2, \dots, a_{2k}\}
   \]

4. **Verification:**
   Calculate the sum \( S \) of the elements in \( A \):
   \[
   S = \sum_{a \in A} a = r + \sum_{i=1}^{2k} a_i
   \]
   Given the construction, we have:
   \[
   S = 2(k-1)S_1
   \]
   We need to show that \( B_i := \{r, a_i, a_{2k+1-i}\} \) for \( i = 1, 2, \dots, k \) are half subsets of \( A \). For each \( i \):
   \[
   r + a_i + a_{2k+1-i} = S/2
   \]
   This means \( B_i \) are indeed half subsets of \( A \).

5. **Conclusion:**
   The number of half subsets is exactly \( 2k \). For \( k = 1 \) and \( k = 2 \), specific examples are:
   - For \( k = 1 \): \( A = \{1, 2, 3\} \) with half subsets \( \{1, 2\} \) and \( \{3\} \).
   - For \( k = 2 \): \( A = \{1, 2, 4, 5\} \) with appropriate half subsets.

The final answer is \( \boxed{ n = 2k } \) for \( k \in \mathbb{Z}^+ \).