Question 3
A function $f$ defined on the positive integers (and taking positive integer values) is given by $$\begin{aligned}f(1)&=1,\quad f(3)=3,\\f(2n)&=f(n),\\f(4n+1)&=2f(2n+1)-f(n),\\f(4n+3)&=3f(2n+1)-2f(n),\end{aligned}$$ for all positive integers $n$. Determine with proof the number of positive integers less than or equal to $1988$ for which $f(n)=n$.