Editing Highly composite number (section)

==Examples==
The first 41 highly composite numbers are listed in the table below  {{OEIS|id=A002182}}. The number of divisors is given in the column labeled ''d''(''n''). Asterisks indicate [[superior highly composite numbers]].

{| class="wikitable" style="text-align:left"
|-
! Order
! HCN<br />''n''
! prime<br /> factorization
! prime<br />exponents
! number<br />of prime<br />factors
! {{abbr|''d''(''n'')|number of divisors of n}}
! primorial<br /> factorization
|-
| 1
| [[1 (number)|1]]
| 
|
| 0
| 1
|
|-
| 2
| [[2 (number)|2]]*
| <math>2</math>
|1
| 1
| 2
| <math>2</math>
|-
| 3
| [[4 (number)|4]]
| <math>2^2</math>
|2
| 2
| 3
| <math>2^2</math>
|-
| 4
| [[6 (number)|6]]*
| <math>2\cdot 3</math>
|1,1
| 2
| 4
| <math>6</math>
|-
| 5
| [[12 (number)|12]]*
| <math>2^2\cdot 3</math>
|2,1
| 3
| 6
| <math>2\cdot 6</math>
|-
| 6
| [[24 (number)|24]]
| <math>2^3\cdot 3</math>
|3,1
| 4
| 8
| <math>2^2\cdot 6</math>
|-
| 7
| [[36 (number)|36]]
| <math>2^2\cdot 3^2</math>
|2,2
| 4
| 9
| <math>6^2</math>
|-
| 8
| [[48 (number)|48]]
| <math>2^4\cdot 3</math>
|4,1
| 5
| 10
| <math>2^3\cdot 6</math>
|-
| 9
| [[60 (number)|60]]*
| <math>2^2\cdot 3\cdot 5</math>
|2,1,1
| 4
| 12
| <math>2\cdot 30</math>
|-
| 10
| [[120 (number)|120]]*
| <math>2^3\cdot 3\cdot 5</math>
|3,1,1
| 5
| 16
| <math>2^2\cdot 30</math>
|-
| 11
| [[180 (number)|180]]
| <math>2^2\cdot 3^2\cdot 5</math>
|2,2,1
| 5
| 18
| <math>6\cdot 30</math>
|-
| 12
| [[240 (number)|240]]
| <math>2^4\cdot 3\cdot 5</math>
|4,1,1
| 6
| 20
| <math>2^3\cdot 30</math>
|-
| 13
| [[360 (number)|360]]*
| <math>2^3\cdot 3^2\cdot 5</math>
|3,2,1
| 6
| 24
| <math>2\cdot 6\cdot 30</math>
|-
| 14
| [[720 (number)|720]]
|<math>2^4\cdot 3^2\cdot 5</math>
|4,2,1
| 7
| 30
| <math>2^2\cdot 6\cdot 30</math>
|-
| 15
| [[840 (number)|840]]
| <math>2^3\cdot 3\cdot 5\cdot 7</math>
|3,1,1,1
| 6
| 32
| <math>2^2\cdot 210</math>
|-
| 16
| 1260
| <math>2^2\cdot 3^2\cdot 5\cdot 7</math>
|2,2,1,1
| 6
| 36
| <math>6\cdot 210</math>
|-
| 17
| 1680
| <math>2^4\cdot 3\cdot 5\cdot 7</math>
|4,1,1,1
| 7
| 40
| <math>2^3\cdot 210</math>
|-
| 18
| [[2520 (number)|2520]]*
| <math>2^3\cdot 3^2\cdot 5\cdot 7</math>
|3,2,1,1
| 7
| 48
| <math>2\cdot 6\cdot 210</math>
|-
| 19
| [[5040 (number)|5040]]*
| <math>2^4\cdot 3^2\cdot 5\cdot 7</math>
|4,2,1,1
| 8
| 60
| <math>2^2\cdot 6\cdot 210</math>
|-
| 20
| 7560
| <math>2^3\cdot 3^3\cdot 5\cdot 7</math>
|3,3,1,1
| 8
| 64
| <math>6^2\cdot 210</math>
|-
| 21
| 10080
| <math>2^5\cdot 3^2\cdot 5\cdot 7</math>
|5,2,1,1
| 9
| 72
| <math>2^3\cdot 6\cdot 210</math>
|-
| 22
| 15120
| <math>2^4\cdot 3^3\cdot 5\cdot 7</math>
|4,3,1,1
| 9
| 80
| <math>2\cdot 6^2\cdot 210</math>
|-
| 23
| 20160
| <math>2^6\cdot 3^2\cdot 5\cdot 7</math>
|6,2,1,1
| 10
| 84
| <math>2^4\cdot 6\cdot 210</math>
|-
| 24
| 25200
| <math>2^4\cdot 3^2\cdot 5^2\cdot 7</math>
|4,2,2,1
| 9
| 90
| <math>2^2\cdot 30\cdot 210</math>
|-
| 25
| 27720
| <math>2^3\cdot 3^2\cdot 5\cdot 7\cdot 11</math>
|3,2,1,1,1
| 8
| 96
| <math>2\cdot 6\cdot 2310</math>
|-
| 26
| 45360
| <math>2^4\cdot 3^4\cdot 5\cdot 7</math>
|4,4,1,1
| 10
| 100
| <math>6^3\cdot 210</math>
|-
| 27
| 50400
| <math>2^5\cdot 3^2\cdot 5^2\cdot 7</math>
|5,2,2,1
| 10
| 108
| <math>2^3\cdot 30\cdot 210</math>
|-
| 28
| 55440*
| <math>2^4\cdot 3^2\cdot 5\cdot 7\cdot 11</math>
|4,2,1,1,1
| 9
| 120
| <math>2^2\cdot 6\cdot 2310</math>
|-
| 29
| 83160
| <math>2^3\cdot 3^3\cdot 5\cdot 7\cdot 11</math>
|3,3,1,1,1
| 9
| 128
| <math>6^2\cdot 2310</math>
|-
| 30
| 110880
| <math>2^5\cdot 3^2\cdot 5\cdot 7\cdot 11</math>
|5,2,1,1,1
| 10
| 144
| <math>2^3\cdot 6\cdot 2310</math>
|-
| 31
| 166320
| <math>2^4\cdot 3^3\cdot 5\cdot 7\cdot 11</math>
|4,3,1,1,1
| 10
| 160
| <math>2\cdot 6^2\cdot 2310</math>
|-
| 32
| 221760
| <math>2^6\cdot 3^2\cdot 5\cdot 7\cdot 11</math>
|6,2,1,1,1
| 11
| 168
| <math>2^4\cdot 6\cdot 2310</math>
|-
| 33
| 277200
| <math>2^4\cdot 3^2\cdot 5^2\cdot 7\cdot 11</math>
|4,2,2,1,1
| 10
| 180
| <math>2^2\cdot 30\cdot 2310</math>
|-
| 34
| 332640
| <math>2^5\cdot 3^3\cdot 5\cdot 7\cdot 11</math>
|5,3,1,1,1
| 11
| 192
| <math>2^2\cdot 6^2\cdot 2310</math>
|-
| 35
| 498960
| <math>2^4\cdot 3^4\cdot 5\cdot 7\cdot 11</math>
|4,4,1,1,1
| 11
| 200
| <math>6^3\cdot 2310</math>
|-
| 36
| 554400
| <math>2^5\cdot 3^2\cdot 5^2\cdot 7\cdot 11</math>
|5,2,2,1,1
| 11
| 216
| <math>2^3\cdot 30\cdot 2310</math>
|-
| 37
| 665280
| <math>2^6\cdot 3^3\cdot 5\cdot 7\cdot 11</math>
|6,3,1,1,1
| 12
| 224
| <math>2^3\cdot 6^2\cdot 2310</math>
|-
| 38
| 720720*
| <math>2^4\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13</math>
|4,2,1,1,1,1
| 10
| 240
| <math>2^2\cdot 6\cdot 30030</math>
|-
| 39
| 1081080
| <math>2^3\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13</math>
|3,3,1,1,1,1
| 10
| 256
| <math>6^2\cdot 30030</math>
|-
| 40
| 1441440*
| <math>2^5\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13</math>
| 5,2,1,1,1,1
| 11
| 288
| <math>2^3\cdot 6\cdot 30030</math>
|-
| 41
| 2162160
| <math>2^4\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13</math>
| 4,3,1,1,1,1
| 11
| 320
| <math>2\cdot 6^2\cdot 30030</math>
|}

The divisors of the first 20 highly composite numbers are shown below.

{| class="wikitable"
! ''n'' !! {{abbr|''d''(''n'')|number of divisors of 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
|-
| 1260 || 36 || 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260
|-
| 1680 || 40 || 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680
|-
| 2520 || 48 || 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520
|-
| 5040 || 60 || 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040
|-
| 7560 || 64 || 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 54, 56, 60, 63, 70, 72, 84, 90, 105, 108, 120, 126, 135, 140, 168, 180, 189, 210, 216, 252, 270, 280, 315, 360, 378, 420, 504, 540, 630, 756, 840, 945, 1080, 1260, 1512, 1890, 2520, 3780, 7560
|}

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

{| class="wikitable" style="text-align:center;table-layout:fixed;"
|-
|colspan="6"| <big>'''The highly composite number: 10080'''</big> <br /> 10080&nbsp;=&nbsp;(2&nbsp;×&nbsp;2&nbsp;×&nbsp;2&nbsp;×&nbsp;2&nbsp;×&nbsp;2) &nbsp;×&nbsp; (3&nbsp;×&nbsp;3) &nbsp;×&nbsp; 5 &nbsp;×&nbsp; 7
|- style="color:#000000;background:#ffffff;"
|style="line-height:1.4" height=64|'''1'''<br />×<br />'''10080'''
|style="line-height:1.4"|  '''2''' <br /> × <br /> ''' 5040'''
|style="line-height:1.4"|     3    <br /> × <br />     3360
|style="line-height:1.4"|  '''4''' <br /> × <br /> ''' 2520'''
|style="line-height:1.4"|     5    <br /> × <br />     2016
|style="line-height:1.4"|  '''6''' <br /> × <br /> ''' 1680'''
|- style="color:#000000;background:#ffffff;"
|style="line-height:1.4" height=64|7<br />× <br />     1440
|style="line-height:1.4"|     8    <br /> × <br /> ''' 1260'''
|style="line-height:1.4"|     9    <br /> × <br />     1120
|style="line-height:1.4"|    10    <br /> × <br />     1008
|style="line-height:1.4"| '''12''' <br /> × <br /> '''  840'''
|style="line-height:1.4"|    14    <br /> × <br /> '''  720'''
|- style="color:#000000;background:#ffffff;"
|style="line-height:1.4" height=64|15<br />×<br />      672
|style="line-height:1.4"|    16    <br /> × <br />      630
|style="line-height:1.4"|    18    <br /> × <br />      560
|style="line-height:1.4"|    20    <br /> × <br />      504
|style="line-height:1.4"|    21    <br /> × <br />      480
|style="line-height:1.4"| '''24''' <br /> × <br />      420
|- style="color:#000000;background:#ffffff;"
|style="line-height:1.4" height=64|28<br />×<br /> '''  360'''
|style="line-height:1.4"|    30    <br /> × <br />      336
|style="line-height:1.4"|    32    <br /> × <br />      315
|style="line-height:1.4"|    35    <br /> × <br />      288
|style="line-height:1.4"| '''36''' <br /> × <br />      280
|style="line-height:1.4"|    40    <br /> × <br />      252
|- style="color:#000000;background:#ffffff;"
|style="line-height:1.4" height=64|42<br />×<br /> '''  240'''
|style="line-height:1.4"|    45    <br /> × <br />      224
|style="line-height:1.4"| '''48''' <br /> × <br />      210
|style="line-height:1.4"|    56    <br /> × <br /> '''  180'''
|style="line-height:1.4"| '''60''' <br /> × <br />      168
|style="line-height:1.4"|    63    <br /> × <br />      160
|- style="color:#000000;background:#ffffff;"
|style="line-height:1.4" height=64|70<br />×<br />      144
|style="line-height:1.4"|    72    <br /> × <br />      140
|style="line-height:1.4"|    80    <br /> × <br />      126
|style="line-height:1.4"|    84    <br /> × <br />  ''' 120'''
|style="line-height:1.4"|    90    <br /> × <br />      112
|style="line-height:1.4"|    96    <br /> × <br />      105
|-
|colspan="6"|'''''Note:&nbsp;''''' Numbers in '''bold''' are themselves '''highly composite numbers'''. <br /> Only the twentieth highly composite number 7560 (=&nbsp;3&nbsp;×&nbsp;2520) is absent.<br />10080 is a so-called [[Smooth number|7-smooth number]] ''{{OEIS|id=A002473}}''.
|}

The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:

: <math>a_0^{14} a_1^9 a_2^6 a_3^4 a_4^4 a_5^3 a_6^3 a_7^3 a_8^2 a_9^2 a_{10}^2 a_{11}^2 a_{12}^2 a_{13}^2 a_{14}^2 a_{15}^2 a_{16}^2 a_{17}^2 a_{18}^{2} a_{19} a_{20} a_{21}\cdots a_{229},</math>

where <math>a_n</math> is the <math>n</math>th successive prime number, and all omitted terms (''a''<sub>22</sub> to ''a''<sub>228</sub>) are factors with exponent equal to one (i.e. the number is <math>2^{14}  \times 3^{9}  \times 5^6  \times \cdots \times 1451</math>). More concisely, it is the product of seven distinct primorials:

: <math>b_0^5 b_1^3 b_2^2 b_4 b_7 b_{18} b_{229},</math>

where <math>b_n</math> is the [[primorial]] <math>a_0a_1\cdots a_n</math>.<ref>{{citation
 | last = Flammenkamp | first = Achim 
 | title = Highly Composite Numbers
 | url = http://wwwhomes.uni-bielefeld.de/achim/highly.html}}.</ref>