I was recently watching a video on applications of the Fibonacci sequence in music, and came across the idea of pisano periods. For some number $n$, the pisano period is the period with which the sequence of Fibonacci numbers taken modulo $n$ repeats. While these numbers seem to grow with $n$, I was curious what would happen if you divided the pisano period by its own modulous.
$$ f(n) = \frac{p(n)}{n} $$
I wrote some code to do this, but it seems like the above function maxes out at 6. For reference:
$$ \begin{align} f(2) = 1.5 \\ f(3) = 2.6 \\ f(5) = 4 \\ f(6) = 4 \\ f(10) = 6 \\ f(50) = 6 \\ f(250) = 6 \\ f(1250) = 6 \\ \dots \\ f(781250) = 6 \end{align} $$
Is there some way to prove the maximum of this function? Is it 6?
On the OEIS, one sees that K. S. Brown showed this. Here is the linked pdf on OEIS related to this fact.