For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$?
1. **Understanding the Problem:**
   We need to find the smallest amusing odd integer greater than 1. A positive integer \( n \) is said to be amusing if there exists a positive integer \( k \) such that \( d(k) = s(k) = n \), where \( d(k) \) is the number of divisors of \( k \) and \( s(k) \) is the digit sum of \( k \).

2. **Analyzing \( d(k) \):**
   Since \( d(k) \) is odd, \( k \) must be a perfect square. This is because the number of divisors of a number is odd if and only if the number is a perfect square. 

3. **Analyzing \( s(k) \):**
   We also need \( s(k) \) to be equal to \( n \). Since \( s(k) \equiv k \pmod{9} \) (a property of digit sums), we have \( n \equiv k \pmod{9} \).

4. **Possible Values of \( k \):**
   Since \( k \) is a perfect square, \( k \equiv 0, 1, 4, 7 \pmod{9} \). We need to find the smallest odd \( n > 1 \) such that \( n \equiv k \pmod{9} \) and \( d(k) = s(k) = n \).

5. **Checking Small Values of \( n \):**
   - For \( n = 3 \): \( k \equiv 3 \pmod{9} \) is not possible since \( k \) must be a perfect square.
   - For \( n = 5 \): \( k \equiv 5 \pmod{9} \) is not possible since \( k \) must be a perfect square.
   - For \( n = 7 \): \( k \equiv 7 \pmod{9} \) is possible. We need \( k \) such that \( d(k) = 7 \) and \( s(k) = 7 \). The form of \( k \) would be \( p^6 \) where \( p \) is a prime. However, checking small primes, \( p^6 \not\equiv 7 \pmod{9} \) for any \( p \).

6. **Next Possible Value \( n = 9 \):**
   - For \( n = 9 \): \( k \equiv 0 \pmod{9} \) is possible. We need \( k \) such that \( d(k) = 9 \) and \( s(k) = 9 \). 
   - Consider \( k = 36 \). We have:
     \[
     d(36) = 9 \quad \text{(since the divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36)}
     \]
     \[
     s(36) = 3 + 6 = 9
     \]
   - Thus, \( k = 36 \) satisfies \( d(k) = s(k) = 9 \).

Conclusion:
The smallest amusing odd integer greater than 1 is \( 9 \).

The final answer is \(\boxed{9}\).