A natural number which has more factors than any number before it. The very composite numbers under one million are: 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720.
Pal Erdos did some work with these numbers, including showing that they are all of the form (pn)a * (pn+1)b * ..., where pn is the nth prime number and a ≥ b ≥ ... ≥ 0. For example:
- 2 = 21: {1, 2}
- 4 = 22: {1, 2, 4}
- 6 = 21 31: {1, 2, 3, 6}
- 12 = 22 31: {1, 2, 3, 4, 6, 12}
- 24 = 23 31: {1, 2, 3, 4, 6, 8, 12, 24}
- 36 = 22 32: {1, 2, 3, 4, 6, 9, 12, 18, 36}
- 48 = 24 31: {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}
- 60 = 22 31 51: {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}
- ...
This sheds some light on the use of numbers such as 60 in various ancient mathematical systems: 60 is the first very composite number divisible by 5, making it convenient as a base and also easy to visualize with a five-fingered hand.