Highly composite number

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On the opposite extreme from a prime number is a highly composite number.

Along with a composite number, a highly composite number is one of two types of number that is the opposite of a prime number. A composite number is the general opposite to a prime. A highly composite number (HCN) is the special opposite of a prime. The set of HCNs is a subset of composite numbers, specifically by being made of the type of composite number that has more factors than any number both lesser than itself and lesser than its higher HCN neighbor.

A prime has the least quantity of factors, namely two. A highly composite number is a number that initiates an increase in quantity of factors over that of any lesser number.

For example, 120 has more factors than any smaller number, and also has more factors than numbers lesser than 180. 180, in turn, has an additional two more factors than 120. And 180 has more factors than numbers less than 240. And 240 has more factors than numbers lesser than 360.

Additionally, the unifying number for all highly composite numbers is 12. This is because of three facts:

  1. the initiation above 6 is always at multiples of twelve but not always at multiples of 6;
  2. the initiation span from 12 to 60 is unbroken; and
  3. 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.

Further reading