MathLabs

Problem 3

A function ff on positive integers is defined by f(1)=1f(1)=1, f(3)=3f(3)=3, f(2n)=f(n)f(2n)=f(n), f(4n+1)=2f(2n+1)−f(n)f(4n+1)=2f(2n+1)-f(n), and f(4n+3)=3f(2n+1)−2f(n)f(4n+3)=3f(2n+1)-2f(n). Determine the number of positive integers n≤1988n\le1988 for which f(n)=nf(n)=n.
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f(n) is odd for every nf(n)\text{ is odd for every }n
Detailed analysis

Induction from the recurrences shows f(n) is always odd. The relation f(2n)=f(n) reduces fixed-point counting to odd representatives.