15. The function $f(k)$ is defined on the set of positive integers $\mathbf{N}^{*}$, is an increasing function, and satisfies $f(f(k))=3k$. Find the value of $f(1)+f(9)+f(96)$.
15. $k \in \mathbf{N}_{+}, f(f(k))=3 k$, then $f[f(f(k))]=f(3 k)$. Also, $f[f(f(k))]=3 f(k)$, so $f(3 k)=3 f(k)$. If $f(1)=1$, then $f(f(1))=3$, i.e., $f(1)=3$, which is a contradiction. Suppose $f(1)=a>1$, since $(f(1))=f(a)=3$, we have $f(1)=2, f(2)=3$, at this point $f(3)=3 f(1)=6, f(6)=3 f(2)=9, f(9)=3 f(3)=18, f(27)=3 f(9)=54, f(54)=3 f(18)=81$. Notice that, as the independent variable changes from 27 to 54, increasing by 27 numbers, the function value also changes from 54 to 81, increasing by 27 numbers. By the monotonicity of $f(x)$, $f(28)=55, f(29)=56, f(30)=57, f(31)=58, f(32)=59, \cdots, f(96)=3 f(32)=177$, thus $f(1)+f(9)+f(96)=2+18+177=197$.