5040 and other Anti-Prime Numbers - Numberphile

Numbers like 24, 60, and 360 are 'highly composite numbers'. Here is a GREAT video on these 'anti-prime' numbers : [this video] These anti-prime numbers were studied first by Ramanujan (of course) and conditions are that: 1) The prime factors must be consecutive (2^n X 3^m not 2^n X 5^m) 2) The powers of the prime factors must be deceasing as the prime number increases (2^4 X 3^3 not 2^1 X 3^5) 3) The numbers end with the the highest prime to the highest prime (2^4 X 3^1 not 2^3 X 3^2). Except for the numbers 2 and 4. More here: [link]

Numberphile made a video about "highly composite numbers" / "anti-primes", which are numbers that have more factors than all smaller numbers. [this video] There is an oeis sequence of them that starts "1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840...", which notably does not include 480. [link]

These numbers are called "Highly Composite Numbers" [0]. Basically, it is a series of numbers where each number has more factors than the number before it (and is the first number with that number of factors). As you hinted, they are especially useful if your number system does not have fractions or decimal places and you still want to divide things. You may recognize the beginning of the series: 1 2 4 6 12 24 36 48 60 120 180 240 360 720 Numberphile [1] calls them "Anti-Primes". [0]: [link] [1]: [this video]

They're all magical because they are small. 12 is a highly factorable number and 10 is not. Ancient Greece was not ignorant of this fact. [link] [this video]