Since $d(k)$ is odd, then $k$ is a perfect square. So $k\equiv0,1,4,7\pmod9$, and we also have $n\equiv s(k)\equiv k\pmod9$. Two smallest possible values for $n$ is 7 and 9. If $n=7$, then $k$ is of the form $p^6$, where $p$ is prime. But it is easy to check that $p^6\not\equiv7\pmod9$ for any $p$. So $n\ge9$. For $k=36$ we get $d(k)=s(k)=9$. Thus the smallest amusing odd integer greater than 1 is 9.