James Grime has “Numberphile”, a series of videos about various numbers and how they relate to more serious bits of maths. He goes outside with a big piece of paper, a marker, and a cameraman, and films himself talking and writing on the paper. (But you don’t have to watch him write! Thanks to the magic of film editing, the numbers appear one by one.) In this one on 220 and 284, “amicable numbers” — the factors of 220 add up to 284, and vice versa. Over at Maths Gear they’re selling pairs of keyrings with these numbers inscribed on them.
James points out that although the (220, 284) pair was known to the ancients, and Euler had found thirty pairs by 1747, it wasn’t until 1866 that B. Nicolò I. Paganini (as far as I know, no relation to the violinist) discovered the pair (1184, 1210). I’m a little surprised by this; you hear the story of great feats of calculation in that era, how did this one slip by? It reminds me of the fact that supposedly people thought 
The whole Numberphile channel is here (currently 19 videos, a couple hours in total). Some of my favorites are:
- 3/4 and Kleiber’s law. The metabolic rate of an animal goes up at the 3/4 power of its mass, perhaps due to metabolic scaling;
- 17 is the minimal number of clues needed to solve a sudoku. On a related note, Jason Rosenhouse and Laura Taalman have a book, Taking Sudoku Seriously
- the initial video on the >number 11, released, somewhat inevitably, on 11/11/11. (If you’re in the UK, that’s 11/11/11.)
God plays dice 