A function
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
defined on the positive integers (and taking positive integers values) is given by:
![\begin{matrix} f(1) = 1, f(3) = 3 \\ f(2n) = f(n) \\ f(4n + 1) = 2f(2n + 1) - f(n) \\ f(4n + 3) = 3f(2n + 1) - 2f(n)\text{,} \end{matrix}](/media/m/a/c/2/ac2090655d8f8ccfde920f97ad99fa33.png)
for all positive integers
![n.](/media/m/4/7/8/478d25bfe04800537ae6e85be9d11ea2.png)
Determine with proof the number of positive integers
![\leq 1988](/media/m/8/c/0/8c0dc02cfaf024da7ead3b6e17fadb3a.png)
for which
%V0
A function $f$ defined on the positive integers (and taking positive integers values) is given by:
$\begin{matrix} f(1) = 1, f(3) = 3 \\ f(2n) = f(n) \\ f(4n + 1) = 2f(2n + 1) - f(n) \\ f(4n + 3) = 3f(2n + 1) - 2f(n)\text{,} \end{matrix}$
for all positive integers $n.$ Determine with proof the number of positive integers $\leq 1988$ for which $f(n) = n.$