Can someone help me express this more concisely?

[Orgasmic Troll]

I was goofing around with sorting reduced fractions, and I ended up focusing on ones less than 1/2, and I was looking at what happens if you have all the reduced fractions less than 1/2 with denominators less than n, where do the fractions with denominator n go .. for example


n = 5, [(x), 1/4, 1/3, (x), 1/2]
n = 6, [(x), 1/5, 1/4, 1/3, 2/5, 1/2]
n = 7, [(x), 1/6, 1/5, 1/4, (x), 1/3, 2/5, (x), 1/2]

the (x)'s are where the new fractions will go

if you look at all the places where you could put a new fraction and put a 1 for where a new fraction will go and a 0 where one doesn't go, you get a binary number for each n

n=3, 1
n=4, 10
n=5, 101
n=6, 1000
n=7, 100101
n=8, 100000100

converting them into decimal, you get the sequence 1, 2, 5, 16, 37, 260, 1089, 8224, 33937, 1048584, 4264225, ...

my questions are..

1) Does anybody even get what I'm talking about?
2) If so, how can I express this more concisely, because I feel like I'm all over the map with my explanation
3) I'm generating these by listing everything out by hand, is there some way I could generate these numbers?

I looked up the sequence on OLEIS and nothing came up, and I think it might be worth submitting, but I don't know how to describe it

Thanks!

[philmoore]

You are looking at what are called Farey sequences, or one half of the Farey sequences. A Farey sequence of order n is an ordered sequence of all fractions between 0 and 1 which have denominator less than or equal to n. Your binary numbers are one way of describing where to insert the new fractions to expand a Farey sequence of order n-1 to order n. Are you finding any interesting patterns? Farey sequences have been studied quite a lot in number theory, and even have an interesting connection to the Riemann hypothesis.

(Note added in edit: The binary expansion for n=6 should read:
n=6, 10000, as is clear in the list of decimal equivalents.)

[Orgasmic Troll]

Originally Posted by philmoore

You are looking at what are called Farey sequences, or one half of the Farey sequences. A Farey sequence of order n is an ordered sequence of all fractions between 0 and 1 which have denominator less than or equal to n. Your binary numbers are one way of describing where to insert the new fractions to expand a Farey sequence of order n-1 to order n. Are you finding any interesting patterns? Farey sequences have been studied quite a lot in number theory, and even have an interesting connection to the Riemann hypothesis.

(Note added in edit: The binary expansion for n=6 should read:
n=6, 10000, as is clear in the list of decimal equivalents.)

Thanks! Nothing is exactly jumping off the page regarding these numbers, but they interest me :)