Find a way to express this sequence $1, \frac{1}{7} , \frac{7}{17} , \frac{7}{23}, \frac{17}{31}, …$

number theoryreal-analysissequences-and-series

I am trying to find a way to express this sequence,

$$ 1, \frac{1}{7}, \frac{7}{17}, \frac{7}{23}, \frac{17}{31}, \frac{1}{41}, \frac{23}{47}, \frac{31}{49}, \frac{17}{73}, \frac{49}{71}, \frac{23}{89}, \frac{7}{103}, … $$

It showed up while searching for solutions to a circle. So far I have found that for any m or n, that is $\frac{m}{n}$, m and n must be apart of the OEIS sequence here.

  1. Numbers whose prime factors are all congruent to +1 or -1 modulo 8.
  2. Numbers of the form $x^2 – 2*y^2$, where x is odd and x and y are relatively prime.

So is there a way to generate this sequence? Are there any interesting properties you noticed looking at it?

(Note: if necessary 1 can be excluded)

Edit: To add some better context the way I generated the sequence was by taking the integer solutions to $x^2+y^2=2(n)^2$ and adding term $\frac{y}{x}$ in reduced form to the sequence if it is new $\forall n$ that came before (thus no two terms in the sequence are the same).

Edit2: My apologies one of the terms was wrongly recorded in the series.

Best Answer

It is possible to generate almost this sequence, except for the value of $1/49$ and the positions of the $1/41$ and $79/119$:

  • find primitive Pythagorean triples, i.e. $a^2+b^2=c^2$ where $a$ and $b$ are coprime and $a \lt b$, and order them by $c$ (and ties perhaps by $a$)
  • calculate $\dfrac{b-a}{b+a}$

You would then get 

a   b   c   (b-a)/(b+a)
0   1   1   1/1
3   4   5   1/7
5   12  13  7/17
8   15  17  7/23
7   24  25  17/31
20  21  29  1/41
12  35  37  23/47
9   40  41  31/49
28  45  53  17/73
11  60  61  49/71
16  63  65  47/79
33  56  65  23/89
48  55  73  7/103
13  84  85  71/97
36  77  85  41/113
39  80  89  41/119
65  72  97  7/137
20  99  101 79/119
60  91  109 31/151 
...

which is close to your sequence.

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