TITLE: Let $f: \mathbb N \rightarrow \mathbb N$ are increasing function such that $f\left(f(n)\right)=3n$. Find $f(2017)$
QUESTION [6 upvotes]: Let $f: \mathbb N \rightarrow \mathbb N$ are increasing function such that 
  $$f\left(f(n)\right)=3n$$
  for any positive integer $n$.
Find $f(2017)$

My work so far:
1) If $m\not=n$ then $f(m)\not= f(n)$
2) $f(3n)=3f(n)$

REPLY [6 votes]: Step I:  if $n=3^a+b$ with $0≤b<3^a$ then $f(n)=2\times 3^a+b$.
Proof:  by induction.  Easy to prove that $f(3^a)=2\times 3^a$ and $f(2\times 3^a)=3^{a+1}$.  As there are exactly $3^a+1$ integers in the interval $\left[2\times 3^a,3^{a+1}\right]$ the claim follows.
Step II.  Remark that $2017=2\times 3^6+559$  
To conclude, we have $f(3^6+559)=2\times 3^6+559=2017$.  thus  $$f(2017)=f(f(3^6+559))=3^7+3\times 559= \fbox {3864}$$
Note:  I would check the arithmetic on this, but the method should be sound.