Recounting the rationals (2008)

I find this incredibly beautiful. Short papers that introduce a "general" audience to a supposedly well-known mathematical idea should really be seen as a valuable contribution in mathematics.

Granted, I am a pure maths PhD. I would be really curious to know how many non-mathematicians can glean value/enjoyment from reading this paper, and what if anything could constitute a necessary/sufficient condition for achieving that.

I still find that the sortest one is this:

take the prime factors of n, the primes with even power go to the numerator with halved powers, the primes with odd powers go to the denominator also with halved powers but rounded up. This is bijective, due to the uniqueness of prime factorization, and of course the numerator and denominator greatest divisor is also guaranteed to be 1, as they not share prime factors at all. So e.g.

45 = 3²5¹-> 3/5 60 = 2²3¹*5¹ -> 2/15 2 -> 1/2, 3 -> 1/3, 4 -> 2/1, 5 -> 1/5, 6 -> 1/6, 7 -> 1/7, 8 -> 1/4, 9 -> 3/1, ...

pdf: https://www.math.upenn.edu/~wilf/website/recounting.pdf