Difference between revisions of "Highly composite numbers"

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(Created page with "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 on...")
 
(this is the earlier page, namely the one to which the prime number page did not link. The prime number page linked only to the composite number page. And the prime number page confounded the term 'highly composite' with 'composite')
 
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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.
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On the opposite extreme from a [[prime]] number is a '''highly composite number'''. A prime has 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, a highly composite number '''''initiates''''' an ''increase'' in quantity of factors over that of any lesser number.
  
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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.
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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.
  
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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.  
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Additionally, the unifying number for all highly composite numbers is 12. This is because of three facts:
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#the ''initiation'' above 6 is always at multiples of twelve but not always at multiples of 6;
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#the initiation span from 12 to 60 is unbroken; and
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#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.
  
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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.
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==Further reading==
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https://danielsmathnotes.wordpress.com/144-and-the-bridge-between-two-mathematical-worlds/
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[[Category:Mathematics]]

Latest revision as of 07:28, January 15, 2017

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

https://danielsmathnotes.wordpress.com/144-and-the-bridge-between-two-mathematical-worlds/

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