Numbers that are not Prime Numbers - they can be evenly divided into small segments
Composite Numbers are numbers that can be divided evenly into smaller segments. This means that they are not Prime Numbers*.
The number 1 is neither Prime nor Composite.
Most numbers are composite numbers. There are an infinite number of Prime Numbers, but a [[Comparing Infinities|Larger Infinite]] Comparing Infinities number of Composite Numbers.
Degrees of Composite-ness
Not all composites are equal.
Highly Composite Number
An integer is considered highly composite if it has more divisors than any integer smaller than it.
| Number | Divisors | Classification |
|---|---|---|
| 1 | 1 | (weird) |
| 2 | 1, 2 | Prime |
| 3 | 1, 3 | Prime |
| 4 | 1, 2, 4 | Highly Composite |
| 5 | 1, 5 | Prime |
| 6 | 1, 2, 3, 6 | Highly Composite |
| 7 | 1, 7 | Prime |
| 8 | 1, 2, 4, 8 | Composite |
| 9 | 1, 3, 9 | Composite |
| 10 | 1, 2, 5, 10 | Composite |
| 11 | 1, 11 | Composite |
| 12 | 1, 2, 3, 4, 6, 12 | Highly Composite |
| 1 is weird. It’s not prime, nor composite. But by the definition of highly composite, it is highly composite. |
The first highly composite numbers are: => syntax here is - {number} - {Cardinality of Divisors}
- 1 - 1
- 2 - 2
- 4 - 3
- 6 - 4
- 12 - 6
- 24 - 8
- 36 - 9
- 48 - 10
- 60 - 12
- 120 - 16
- 180 - 18
- 240 - 20
- 360 - 24
- 720 - 30
- 840 - 32
- 1260 - 36
- 1680 - 40
- 2520 - 48
- 5040 - 60
- 7560 - 64
- 10080 - 72
Example - Divisors of 360
360 has 24 divisors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360
Highly Composite numbers are why we have 24 hours in a day, separated into 12 hour halves. Why we say a circle has 360 degrees.
Superior Highly Composite Numbers
An integer is considered superior highly composite if… actually the definition of this gets into the realm of weird number classifications that don’t seem useful. It has to do with raising to partial exponents and comparing to all other numbers. Suffice to say: 1, 6, 12, 60, 120, 360, 2520, 5040, 55440, and 720720 are the first 10 superior highly composite numbers.
Source
- Wikipedia & Wolfram|Alpha