Is there a way to find the $n$th term of a Farey sequence? I was doing some reading on Farey Fractions and was curious if there is a method to find the $n$th term in a particular Farey sequence? I know you could do this with a computer search, but at large denominator values this quickly becomes tedious.
Here's a random example: Find the $81$st term of the Farey sequence with denominator 1500.
 A: Have you seen the next term section of the wikipedia page? They show how to compute the $k+2$th term in the $n$th Farey sequence using the $k$th and $k+1$st terms, and provide python code to actually do it!
If you're interested in very large denominators, Pawlewicz's Order Statistics in the Farey Sequences in Sublinear Time (available here) gives an
$O\left (n^{3/4}\log(n) \right )$ algorithm for computing the $k$th entry in the $n$th Farey sequence. This is much better than wikipedia's algorithm, so if you implement the work in the paper, you stand to gain a substantial amount of speed.
Since so much effort has been put into finding efficient algorithms for computing these numbers, and some cursory googling hasn't turned up a closed form, I'm skeptical that a closed form exists, so these are probably the best we can do.
Edit:
I forgot about your example! The python code on wikipedia takes roughtly $1$ second to compute the entirety of the $F_{1500}$, and from there it's easy to read off the $81$st entry ($0$-indexed) to be $1/1420$.
If we're slightly smarter, and only compute the first $82$ entries instead of all $684183$, it only takes roughly $100$ microseconds (of course, this timesave goes away when we start asking for entries nearer the end of $F_{1500}$).

I hope this helps ^_^
