An Elegant function with elegant conditions

Algebra Level 5

\[\begin {cases} f(1) =1 \\f(3) =3 \\f(2n) = f(n)\\ f(4n+1) =2f(2n+1)-f(n)\\ f(4n+3)=3f(2n+1)-2f(n) \end {cases} \] The function \(f\) is defined on the set of all positive integers as above. Find the number of \(n\) with \(f(n)=n\) for \(1\leq n\leq 1988\).

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