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 ab ≥ ... ≥ 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.

The property discovered by Pal Erdos alluded to in the above writeup is an easy consequence of the definition. In fact, we can show that any very composite number is of the form

N = 2k13k25k3...
where k1≥k2≥...

Indeed, for any indices a < b, we have that

N' = 2k13k2...pakb...pbka...
(i.e. what you get by switching the powers if pa and pb in N) has exactly the same number of factors as does N. It follows from the definition that (if N is very composite) N ≤ N', so ka ≥ kb, as required.

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