When $n\geq2$, let $a_n = \lceil \frac{n}{\pi}\rceil$ and let $b_n = \lceil{\csc({\frac{\pi}{n}})}\rceil$.

Question

When $n\geq2$, let $a_n =\left \lceil \frac{n}{\pi}\right\rceil$ and let $b_n = \left\lceil{\csc({\frac{\pi}{n}})}\right\rceil$.

The terms of the sequences starting with $n = 2$ are:

{$a_n$} = $1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, ...$ and

{$b_n$} = $1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, ...$

Note that the sequences differ when $n=3$. Is it true that $a_n = b_n$ for all $n>3$?

Answer 1

No. When $n = 80143857$, $$\begin{align} n / \pi &= 25510581.9999999952976568107626972575226719258409876014\cdots, \\ \csc(\pi/n) &= 25510582.0000000018308933581165258478077895828199645771\cdots. \end{align}$$ This counterexample was found with a small JavaScript code and verified by Wolfram|Alpha.