Problem Archive

The function $f$ is defined for all positive integers as follows:

• $f(1)=1$
• $f(3)=3$
• $f(2n)=f(n)$
• $f(4n + 1)=2f(2n + 1) - f(n)$
• $f(4n + 3)=3f(2n + 1) - 2f(n)$

The function $S(n)$ is defined as $\sum_{i=1}^{n}f(i)$.

$S(8)=22$ and $S(100)=3604$.

Find $S(3^{37})$. Give the last $9$ digits of your answer.