# Difference between revisions of "Highly composite numbers"

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− | On the opposite extreme from the primes are the highly composite numbers. The primes have the least quantity of factors, namely two. The highly composite numbers are those not only with the greatest quantity of factors compared to all lesser numbers, but the quantity of factors of which remains un-exceeded by any greater number its vicinity. | + | On the opposite extreme from the primes are the highly composite numbers. The primes have the least quantity of factors, namely two. The highly composite numbers are those not only with the greatest quantity of factors compared to all lesser numbers, but the quantity of factors of which remains un-exceeded by any greater number in its vicinity. |

For example, every number above 180 has less factors than has 180, until you get to 240. Then, 240 has an additional two factors over 180. Then, no number above 240 has more factors than 240, until you get to 360, while the quantity of factors of 336 mere ''equals'' that of 240. | For example, every number above 180 has less factors than has 180, until you get to 240. Then, 240 has an additional two factors over 180. Then, no number above 240 has more factors than 240, until you get to 360, while the quantity of factors of 336 mere ''equals'' that of 240. |

## Revision as of 22:01, 17 December 2013

On the opposite extreme from the primes are the highly composite numbers. The primes have the least quantity of factors, namely two. The highly composite numbers are those not only with the greatest quantity of factors compared to all lesser numbers, but the quantity of factors of which remains un-exceeded by any greater number in its vicinity.

For example, every number above 180 has less factors than has 180, until you get to 240. Then, 240 has an additional two factors over 180. Then, no number above 240 has more factors than 240, until you get to 360, while the quantity of factors of 336 mere *equals* that of 240.

So, the way to identify a number as highly composite is if it * initiates* an increase in quantity of factors above that of all lesser numbers.

The key factor for all highly composite numbers beginning with 12 is 12. So, every highly composite number greater than 12 is divisible by twelve. And, otherwise, there is no single number (except, of course, 1) by which all highly composite numbers are divisible.