Let $f_n$ denote the $n$th Fibonacci number. A positive integer $n$ is called good if $f_{f_n}$ is divisible by $n$ but $f_n$ is not divisible by $n$.
My question is: how many good numbers are there. I think there are infinity many but I can’t prove it. Thanks!
Anyway this is from a russian book called: All about Fibonacci
Suppose $n$ is good. Since $\gcd(f_m,f_n)=f_{\gcd(m,n)}$, we have $$\gcd(f_{f_{f_n}},f_n)=f_{\gcd(f_{f_n},n)}=f_n,$$ and $$\gcd(f_{f_n},f_n)=f_{\gcd(f_n,n)}<f_n.$$ Therefore, $f_n$ is good. So if there is one good number, there must exist infinitely many good numbers.