Question 21
Let $S = \{1, 2, \dots, 100\}$. Determine the number of functions $f : S \to S$ satisfying the following conditions.
(i) $f(1) = 1$;
(ii) $f$ is bijective (i.e. for every $y$ in $S$, the equation $f(x) = y$ has exactly one solution);
(iii) $f(n) = f(g(n))f(h(n))$ for every $n$ in $S$.
Here $g(n)$ and $h(n)$ denote the uniquely determined positive integers such that $g(n) \le h(n)$, $g(n)h(n) = n$ and $h(n)-g(n)$ is as small as possible. (For instance, $g(80) = 8$, $h(80) = 10$ and $g(81) = h(81) = 9$.)
(i) $f(1) = 1$;
(ii) $f$ is bijective (i.e. for every $y$ in $S$, the equation $f(x) = y$ has exactly one solution);
(iii) $f(n) = f(g(n))f(h(n))$ for every $n$ in $S$.
Here $g(n)$ and $h(n)$ denote the uniquely determined positive integers such that $g(n) \le h(n)$, $g(n)h(n) = n$ and $h(n)-g(n)$ is as small as possible. (For instance, $g(80) = 8$, $h(80) = 10$ and $g(81) = h(81) = 9$.)