Powers of 2 and 3

1	3	9	27	81	243	729	2187	6561	19683	59049
2	6	18	54
4	12	36	108	324
8	24	72	216
16	48	144	432	1296
32	96	288			7776
64	192	576	1728			46656
128	384	1152
256	768	2304
512	1536
1024	3072
2048	6144
4096	12288
8192	24576
16384	49152
32768
65536

It's not a complete table because I've only included numbers that I recognize. The first three columns I know mainly from Apple II days; and of course the main diagonal I know from working out dice probabilities. I'd like to include 3456 (below 1728) and 3888 (above 7776), just because they're such nice numbers, but I have to admit I wouldn't have recognized them.

I'll also admit to having favorites. I like all the small numbers, but there familiarity breeds taking for granted. I like 243, partly because it reminds me of 7³ = 343; I also like 59049 a lot. Then there's 216, which is very pleasing for some reason.

Among the larger powers of two, of course I'm very fond of 256, and also of 65536, which was the size of the Apple's address space. Speaking of which, some numbers are very familiar because they were nice round addresses in hexadecimal: 768 (300) and 24576 (6000). I don't think of 192 (C0) as an address, but it's also a nice round number, and very handy.

Now, here's an interesting thing I never noticed before. I was going to say that I liked how there happened to be some pairs of similar numbers, like 288 and 12288, or 576 and 24576. But, it turns out that's not just a coincidence, it actually follows from the useful equation

1024 = 1000 + 24.

If you multiply that by any entry N from the table, and rearrange a little, you get

2¹⁰ N = 1000 N + 2³ 3 N,

which has a nice translation into English: the entry ten spaces down from N is equal to a thousand times N plus the entry three spaces down and one over.

What's more, you can do the same thing for any equation that contains only powers of 2, 3, and 5. For example, there's a rule based on the Pythagorean triangle.

3² + 4² = 5²