Highly composite number

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A **highly composite number** in math (also called _anti-prime_) is a real number with more divisors than any real number smaller than it.

Jean-Pierre Kahane thought that Plato might have known about highly composite numbers. This is because he chose 5040 as a good number of citizens in a city as 5040 has more divisors than any numbers less than it.

## Examples

The first 38 highly composite numbers are listed in the table below  . The number of divisors is given in the column labeled _d_(_n_). The letters with asterisks are also superior highly composite numbers.

|Order|HCN _n_|prime factorization|prime exponents|number of prime factors||primorial factorization|
| --- | --- | --- | --- | --- | --- | --- |
|1|1|||0|1||
|2|2*||1|1|2||
|3|4||2|2|3||
|4|6*||1,1|2|4||
|5|12*||2,1|3|6||
|6|24||3,1|4|8||
|7|36||2,2|4|9||
|8|48||4,1|5|10||
|9|60*||2,1,1|4|12||
|10|120*||3,1,1|5|16||
|11|180||2,2,1|5|18||
|12|240||4,1,1|6|20||
|13|360*||3,2,1|6|24||
|14|720||4,2,1|7|30||
|15|840||3,1,1,1|6|32||
|16|1260||2,2,1,1|6|36||
|17|1680||4,1,1,1|7|40||
|18|2520*||3,2,1,1|7|48||
|19|5040*||4,2,1,1|8|60||
|20|7560||3,3,1,1|8|64||
|21|10080||5,2,1,1|9|72||
|22|15120||4,3,1,1|9|80||
|23|20160||6,2,1,1|10|84||
|24|25200||4,2,2,1|9|90||
|25|27720||3,2,1,1,1|8|96||
|26|45360||4,4,1,1|10|100||
|27|50400||5,2,2,1|10|108||
|28|55440*||4,2,1,1,1|9|120||
|29|83160||3,3,1,1,1|9|128||
|30|110880||5,2,1,1,1|10|144||
|31|166320||4,3,1,1,1|10|160||
|32|221760||6,2,1,1,1|11|168||
|33|277200||4,2,2,1,1|10|180||
|34|332640||5,3,1,1,1|11|192||
|35|498960||4,4,1,1,1|11|200||
|36|554400||5,2,2,1,1|11|216||
|37|665280||6,3,1,1,1|12|224||
|38|720720*||4,2,1,1,1,1|10|240||

The divisor of the first 15 highly composite numbers are shown below.

|_n_||Divisors of _n_|
| --- | --- | --- |
|1|1|1|
|2|2|1, 2|
|4|3|1, 2, 4|
|6|4|1, 2, 3, 6|
|12|6|1, 2, 3, 4, 6, 12|
|24|8|1, 2, 3, 4, 6, 8, 12, 24|
|36|9|1, 2, 3, 4, 6, 9, 12, 18, 36|
|48|10|1, 2, 3, 4, 6, 8, 12, 16, 24, 48|
|60|12|1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60|
|120|16|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120|
|180|18|1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180|
|240|20|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240|
|360|24|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360|
|720|30|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720|
|840|32|1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840|

The table below shows all 72 divisors of 10080 by writing it as a product of two numbers in 36 different ways.

|colspan="6"|<big>**The highly composite number: 10080**</big> 10080 = (2 × 2 × 2 × 2 × 2)  ×  (3 × 3)  ×  5  ×  7|||||||||||
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|style="line-height:1.4" height=64|**1** × **10080**|style="line-height:1.4"|**2** × ** 5040**|style="line-height:1.4"|3 × 3360|style="line-height:1.4"|**4** × ** 2520**|style="line-height:1.4"|5 × 2016|style="line-height:1.4"|**6** × ** 1680**|
|style="line-height:1.4" height=64|7 × 1440|style="line-height:1.4"|8 × ** 1260**|style="line-height:1.4"|9 × 1120|style="line-height:1.4"|10 × 1008|style="line-height:1.4"|**12** × **  840**|style="line-height:1.4"|14 × **  720**|
|style="line-height:1.4" height=64|15 × 672|style="line-height:1.4"|16 × 630|style="line-height:1.4"|18 × 560|style="line-height:1.4"|20 × 504|style="line-height:1.4"|21 × 480|style="line-height:1.4"|**24** × 420|
|style="line-height:1.4" height=64|28 × **  360**|style="line-height:1.4"|30 × 336|style="line-height:1.4"|32 × 315|style="line-height:1.4"|35 × 288|style="line-height:1.4"|**36** × 280|style="line-height:1.4"|40 × 252|
|style="line-height:1.4" height=64|42 × **  240**|style="line-height:1.4"|45 × 224|style="line-height:1.4"|**48** × 210|style="line-height:1.4"|56 × **  180**|style="line-height:1.4"|**60** × 168|style="line-height:1.4"|63 × 160|
|style="line-height:1.4" height=64|70 × 144|style="line-height:1.4"|72 × 140|style="line-height:1.4"|80 × 126|style="line-height:1.4"|84 × ** 120**|style="line-height:1.4"|90 × 112|style="line-height:1.4"|96 × 105|
|colspan="6"|***Note: *** The numbers in **bold** are also **highly composite numbers**. 10080 is often referred to as a 7-smooth number __.|||||||||||

## Similar sequences

Every highly composite number that is bigger than 6 is also an abundant number. Not all highly composite numbers are also Harshad numbers, however most of them are the same. The first highly composite number that is not a Harshad number is 245,044,800. This number's digit's sum is 27. 27, however, doesn't divide into 245,044,800 evenly.

10 of the first 38 highly composite numbers are also superior highly composite numbers.

## Related pages

- Abundant number
- Euler's totient function
- Harshad number

## References

- Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.

## Other websites

- Algorithm for computing Highly Composite Numbers
- First 10000 Highly Composite Numbers as factors
- Achim Flammenkamp, First 779674 HCN with sigma,tau,factors
- Online Highly Composite Numbers Calculator

- 5040 and other Anti-Prime Numbers - Dr. James Grime by Dr. James Grime for Numberphile