3

5

7

11

13

19

23

29

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I just saw this in Richard Guys "The Strong Law of Small Numbers" (which I was looking at via @AlexKontorovich and @lpachter).

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Also sadly, I don't have a very good answer for what's going on. There's a nice partial answer, but not one that leaves us with much satisfaction for what primes are doing here, other than to say "coincidence".

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The rule for generating new rows above is just a cute fact arising from how the fraction sitting between (a/b) and (c/d) in such a sequence looks like (a+c)/(b+d).

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0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1.

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(Also, fun fact, the length of a Farey sequence asymptotically approaches 3*n^2 / pi^2)

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There can only be so many patterns among small numbers, so some are bound to collide.

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