I've been reading all my books that include some Number Theory, and doing some Study of Numbers, which I'm thinking of publishing under the title "Numerology: The Wisdom and Folly of Numbers".
Here is a little item that is probably not new, but I can't find it in any of my sources. Can anyone locate it in some publication? I've a vague idea I've seen something like it somewhere.
I call numbers of the form (2^r)x(3^s) "Basals". In other words they are any numbers that exclude prime factors greater than 3. .The sequence runs: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 98, 108, 128, ...
The corresponding powers of 2 and 3 in the sequence run: (0,0), (1,0), (0,1), (2,0), (1,1), (3,0), (0,2), (2,1), (4,0) and so on. Every pairing occurs and they are listed in a unique order.
This scheme thus provides an enumeration of the grid cells of an endless board, as partially illustrated in this tour diagram, from (0,0) = 1 to (9,0) = 512.
It is interesting that the path appears never to cross itself.
Further: This diagram shows the similar result obtained using (2^r)x(5^s) to determine the sequence. This goes up to 5^4 = 625 at (0,4). The moves in the upward direction tend to be knight moves.