Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.
To solve the problem, we need to find all positive integers \( n \) such that \( n + d(n) = d(n)^2 \), where \( d(n) \) is the number of positive divisors of \( n \).

1. **Understanding the equation:**
   \[
   n + d(n) = d(n)^2
   \]
   Rearrange the equation to:
   \[
   n = d(n)^2 - d(n)
   \]
   Let \( k = d(n) \). Then:
   \[
   n = k(k - 1)
   \]

2. **Finding possible values of \( k \):**
   We need to find \( k \) such that \( n = k(k - 1) \) has exactly \( k \) divisors. We will test various values of \( k \) and check if \( n \) has exactly \( k \) divisors.

3. **Testing small values of \( k \):**
   - \( k = 1 \):
     \[
     n = 1 \cdot (1 - 1) = 0 \quad \text{(not a positive integer)}
     \]
   - \( k = 2 \):
     \[
     n = 2 \cdot (2 - 1) = 2
     \]
     Check \( d(2) \):
     \[
     d(2) = 2 \quad \text{(correct)}
     \]
   - \( k = 3 \):
     \[
     n = 3 \cdot (3 - 1) = 6
     \]
     Check \( d(6) \):
     \[
     d(6) = 4 \quad \text{(incorrect)}
     \]
   - \( k = 4 \):
     \[
     n = 4 \cdot (4 - 1) = 12
     \]
     Check \( d(12) \):
     \[
     d(12) = 6 \quad \text{(incorrect)}
     \]
   - \( k = 5 \):
     \[
     n = 5 \cdot (5 - 1) = 20
     \]
     Check \( d(20) \):
     \[
     d(20) = 6 \quad \text{(incorrect)}
     \]
   - \( k = 6 \):
     \[
     n = 6 \cdot (6 - 1) = 30
     \]
     Check \( d(30) \):
     \[
     d(30) = 8 \quad \text{(incorrect)}
     \]
   - \( k = 7 \):
     \[
     n = 7 \cdot (7 - 1) = 42
     \]
     Check \( d(42) \):
     \[
     d(42) = 8 \quad \text{(incorrect)}
     \]
   - \( k = 8 \):
     \[
     n = 8 \cdot (8 - 1) = 56
     \]
     Check \( d(56) \):
     \[
     d(56) = 8 \quad \text{(correct)}
     \]
   - \( k = 9 \):
     \[
     n = 9 \cdot (9 - 1) = 72
     \]
     Check \( d(72) \):
     \[
     d(72) = 12 \quad \text{(incorrect)}
     \]
   - \( k = 10 \):
     \[
     n = 10 \cdot (10 - 1) = 90
     \]
     Check \( d(90) \):
     \[
     d(90) = 12 \quad \text{(incorrect)}
     \]
   - \( k = 12 \):
     \[
     n = 12 \cdot (12 - 1) = 132
     \]
     Check \( d(132) \):
     \[
     d(132) = 12 \quad \text{(correct)}
     \]
   - \( k = 13 \):
     \[
     n = 13 \cdot (13 - 1) = 156
     \]
     Check \( d(156) \):
     \[
     d(156) = 12 \quad \text{(incorrect)}
     \]
   - \( k = 15 \):
     \[
     n = 15 \cdot (15 - 1) = 210
     \]
     Check \( d(210) \):
     \[
     d(210) = 16 \quad \text{(incorrect)}
     \]
   - \( k = 16 \):
     \[
     n = 16 \cdot (16 - 1) = 240
     \]
     Check \( d(240) \):
     \[
     d(240) = 20 \quad \text{(incorrect)}
     \]
   - \( k = 21 \):
     \[
     n = 21 \cdot (21 - 1) = 420
     \]
     Check \( d(420) \):
     \[
     d(420) = 24 \quad \text{(incorrect)}
     \]
   - \( k = 24 \):
     \[
     n = 24 \cdot (24 - 1) = 552
     \]
     Check \( d(552) \):
     \[
     d(552) = 16 \quad \text{(incorrect)}
     \]
   - \( k = 25 \):
     \[
     n = 25 \cdot (25 - 1) = 600
     \]
     Check \( d(600) \):
     \[
     d(600) = 24 \quad \text{(incorrect)}
     \]
   - \( k = 36 \):
     \[
     n = 36 \cdot (36 - 1) = 1260
     \]
     Check \( d(1260) \):
     \[
     d(1260) = 36 \quad \text{(correct)}
     \]

4. **Conclusion:**
   The possible values of \( n \) that satisfy the equation \( n + d(n) = d(n)^2 \) are \( n = 2, 56, 132, 1260 \).

The final answer is \( \boxed{2, 56, 132, 1260} \).