### More on the Above "Proof"

The above "proof" adduces the square-free integers--a very interesting set. For all finite sets P of primes, Product P ordered by Sum P (and within each tranche by Size P) give a canonical ordering of the square-free integers:

2, 3, 5, 6, 7, 10, 15, 14, 21, 30, 11, 35, 42, 13, 22, 33, 70, 26, 105, 39, 55, 66, 17, 210, 65, 77, 78, 110, 19, 34, 165, 51, 91, 130, 154, 38, 195, 231, 330, 57, 85, 102, 182, 23, 273, 385, 390, 462, 95, 119, 143, 114, 170, 46, 255, 455, 546, 770, 69, 133, 190, 238, 286, 1155, 285, 357, 429, 510, 910, 115, 187, 138, 266, 1365, 29, 399, 595, 715, 570, 714, 858, ...

This uncomfortable order reflects the uncomfortable properties of primes in general, which is of course why I like it.

Like the primes themselves, the square-free integers may be isolated by the Sieve of Eratosthenes thus:

1) Write down the integers greater than 1.
3) Strike it and all multiples.
4) Take the next square.
5) If it remains, strike it and all remaining multiples.
6) Goto (4).