Highly composite numbers
On the opposite extreme from the primes are the highly composite numbers. The primes have the least quantity of factors, namely two. A highly composite number is one which has a greater number of factors than any smaller number. In other words, the highly composite numbers each initiate an increase in quantity of factors over that of any lesser number.
For example, 180 has more factors than any smaller number, or any greater one up to 240. Then, 240 has an additional two more factors than 180. Then, no number above 240 has more factors than 240, until you get to 360, while the quantity of factors of 336 merely equals that of 240.
Additionally, the unifying number for all highly composite numbers is 12. This is because of three facts:
- the initiation above 6 is always at multiples of twelve but not always at multiples of 6;
- the initiation span from 12 to 60 is unbroken; and
- composite equivalence is unbroken from 60 to 120. These three facts together imply a spike graph beginning with 1 and ending with 120, in which there are two sequences of six spikes, in which the 60 spike functions both as the final of the first sequence and as the initial of the second sequence, and the second sequence for which only its first and final numbers are initials.