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Highly composite number
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<h2 id="examples">Examples</h2> <p>The first 41 highly composite numbers are listed in the table below (sequence A002182 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>). The number of divisors is given in the column labeled <i>d</i>(<i>n</i>). Asterisks indicate <a href="/facts/Superior_highly_composite_numbers/NzD4z9QY">superior highly composite numbers</a>. </p> <table><tbody><tr><th>Order</th><th>HCN<i>n</i></th><th>prime factorization</th><th>primeexponents</th><th>numberof primefactors</th><th><i>d</i>(<i>n</i>)</th><th>primorial factorization</th></tr><tr><td>1</td><td><a href="/facts/1_(number)/m3wpSfMf">1</a></td><td></td><td></td><td>0</td><td>1</td><td></td></tr><tr><td>2</td><td><a href="/facts/2_(number)/u5WC44t9">2</a>*</td><td> 2 {\displaystyle 2} </td><td>1</td><td>1</td><td>2</td><td> 2 {\displaystyle 2} </td></tr><tr><td>3</td><td><a href="/facts/4_(number)/zwN4alXI">4</a></td><td> 2 2 {\displaystyle 2^{2}} </td><td>2</td><td>2</td><td>3</td><td> 2 2 {\displaystyle 2^{2}} </td></tr><tr><td>4</td><td><a href="/facts/6_(number)/MsEd6F54">6</a>*</td><td> 2 ⋅ 3 {\displaystyle 2\cdot 3} </td><td>1,1</td><td>2</td><td>4</td><td> 6 {\displaystyle 6} </td></tr><tr><td>5</td><td><a href="/facts/12_(number)/j8162cvV">12</a>*</td><td> 2 2 ⋅ 3 {\displaystyle 2^{2}\cdot 3} </td><td>2,1</td><td>3</td><td>6</td><td> 2 ⋅ 6 {\displaystyle 2\cdot 6} </td></tr><tr><td>6</td><td><a href="/facts/24_(number)/YKtxDTGm">24</a></td><td> 2 3 ⋅ 3 {\displaystyle 2^{3}\cdot 3} </td><td>3,1</td><td>4</td><td>8</td><td> 2 2 ⋅ 6 {\displaystyle 2^{2}\cdot 6} </td></tr><tr><td>7</td><td><a href="/facts/36_(number)/699GXfY8">36</a></td><td> 2 2 ⋅ 3 2 {\displaystyle 2^{2}\cdot 3^{2}} </td><td>2,2</td><td>4</td><td>9</td><td> 6 2 {\displaystyle 6^{2}} </td></tr><tr><td>8</td><td><a href="/facts/48_(number)/jRerKrTP">48</a></td><td> 2 4 ⋅ 3 {\displaystyle 2^{4}\cdot 3} </td><td>4,1</td><td>5</td><td>10</td><td> 2 3 ⋅ 6 {\displaystyle 2^{3}\cdot 6} </td></tr><tr><td>9</td><td><a href="/facts/60_(number)/EawDWoBh">60</a>*</td><td> 2 2 ⋅ 3 ⋅ 5 {\displaystyle 2^{2}\cdot 3\cdot 5} </td><td>2,1,1</td><td>4</td><td>12</td><td> 2 ⋅ 30 {\displaystyle 2\cdot 30} </td></tr><tr><td>10</td><td><a href="/facts/120_(number)/Q6ceL3CM">120</a>*</td><td> 2 3 ⋅ 3 ⋅ 5 {\displaystyle 2^{3}\cdot 3\cdot 5} </td><td>3,1,1</td><td>5</td><td>16</td><td> 2 2 ⋅ 30 {\displaystyle 2^{2}\cdot 30} </td></tr><tr><td>11</td><td><a href="/facts/180_(number)/3ceu1HSd">180</a></td><td> 2 2 ⋅ 3 2 ⋅ 5 {\displaystyle 2^{2}\cdot 3^{2}\cdot 5} </td><td>2,2,1</td><td>5</td><td>18</td><td> 6 ⋅ 30 {\displaystyle 6\cdot 30} </td></tr><tr><td>12</td><td><a href="/facts/240_(number)/XXMIa4AK">240</a></td><td> 2 4 ⋅ 3 ⋅ 5 {\displaystyle 2^{4}\cdot 3\cdot 5} </td><td>4,1,1</td><td>6</td><td>20</td><td> 2 3 ⋅ 30 {\displaystyle 2^{3}\cdot 30} </td></tr><tr><td>13</td><td><a href="/facts/360_(number)/vvTdIepn">360</a>*</td><td> 2 3 ⋅ 3 2 ⋅ 5 {\displaystyle 2^{3}\cdot 3^{2}\cdot 5} </td><td>3,2,1</td><td>6</td><td>24</td><td> 2 ⋅ 6 ⋅ 30 {\displaystyle 2\cdot 6\cdot 30} </td></tr><tr><td>14</td><td><a href="/facts/720_(number)/btpCIXRe">720</a></td><td> 2 4 ⋅ 3 2 ⋅ 5 {\displaystyle 2^{4}\cdot 3^{2}\cdot 5} </td><td>4,2,1</td><td>7</td><td>30</td><td> 2 2 ⋅ 6 ⋅ 30 {\displaystyle 2^{2}\cdot 6\cdot 30} </td></tr><tr><td>15</td><td><a href="/facts/840_(number)/Lpupvl05">840</a></td><td> 2 3 ⋅ 3 ⋅ 5 ⋅ 7 {\displaystyle 2^{3}\cdot 3\cdot 5\cdot 7} </td><td>3,1,1,1</td><td>6</td><td>32</td><td> 2 2 ⋅ 210 {\displaystyle 2^{2}\cdot 210} </td></tr><tr><td>16</td><td>1260</td><td> 2 2 ⋅ 3 2 ⋅ 5 ⋅ 7 {\displaystyle 2^{2}\cdot 3^{2}\cdot 5\cdot 7} </td><td>2,2,1,1</td><td>6</td><td>36</td><td> 6 ⋅ 210 {\displaystyle 6\cdot 210} </td></tr><tr><td>17</td><td>1680</td><td> 2 4 ⋅ 3 ⋅ 5 ⋅ 7 {\displaystyle 2^{4}\cdot 3\cdot 5\cdot 7} </td><td>4,1,1,1</td><td>7</td><td>40</td><td> 2 3 ⋅ 210 {\displaystyle 2^{3}\cdot 210} </td></tr><tr><td>18</td><td><a href="/facts/2520_(number)/DpqUYGng">2520</a>*</td><td> 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 {\displaystyle 2^{3}\cdot 3^{2}\cdot 5\cdot 7} </td><td>3,2,1,1</td><td>7</td><td>48</td><td> 2 ⋅ 6 ⋅ 210 {\displaystyle 2\cdot 6\cdot 210} </td></tr><tr><td>19</td><td><a href="/facts/5040_(number)/BMhEgbL9">5040</a>*</td><td> 2 4 ⋅ 3 2 ⋅ 5 ⋅ 7 {\displaystyle 2^{4}\cdot 3^{2}\cdot 5\cdot 7} </td><td>4,2,1,1</td><td>8</td><td>60</td><td> 2 2 ⋅ 6 ⋅ 210 {\displaystyle 2^{2}\cdot 6\cdot 210} </td></tr><tr><td>20</td><td>7560</td><td> 2 3 ⋅ 3 3 ⋅ 5 ⋅ 7 {\displaystyle 2^{3}\cdot 3^{3}\cdot 5\cdot 7} </td><td>3,3,1,1</td><td>8</td><td>64</td><td> 6 2 ⋅ 210 {\displaystyle 6^{2}\cdot 210} </td></tr><tr><td>21</td><td>10080</td><td> 2 5 ⋅ 3 2 ⋅ 5 ⋅ 7 {\displaystyle 2^{5}\cdot 3^{2}\cdot 5\cdot 7} </td><td>5,2,1,1</td><td>9</td><td>72</td><td> 2 3 ⋅ 6 ⋅ 210 {\displaystyle 2^{3}\cdot 6\cdot 210} </td></tr><tr><td>22</td><td>15120</td><td> 2 4 ⋅ 3 3 ⋅ 5 ⋅ 7 {\displaystyle 2^{4}\cdot 3^{3}\cdot 5\cdot 7} </td><td>4,3,1,1</td><td>9</td><td>80</td><td> 2 ⋅ 6 2 ⋅ 210 {\displaystyle 2\cdot 6^{2}\cdot 210} </td></tr><tr><td>23</td><td>20160</td><td> 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 {\displaystyle 2^{6}\cdot 3^{2}\cdot 5\cdot 7} </td><td>6,2,1,1</td><td>10</td><td>84</td><td> 2 4 ⋅ 6 ⋅ 210 {\displaystyle 2^{4}\cdot 6\cdot 210} </td></tr><tr><td>24</td><td>25200</td><td> 2 4 ⋅ 3 2 ⋅ 5 2 ⋅ 7 {\displaystyle 2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7} </td><td>4,2,2,1</td><td>9</td><td>90</td><td> 2 2 ⋅ 30 ⋅ 210 {\displaystyle 2^{2}\cdot 30\cdot 210} </td></tr><tr><td>25</td><td>27720</td><td> 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{3}\cdot 3^{2}\cdot 5\cdot 7\cdot 11} </td><td>3,2,1,1,1</td><td>8</td><td>96</td><td> 2 ⋅ 6 ⋅ 2310 {\displaystyle 2\cdot 6\cdot 2310} </td></tr><tr><td>26</td><td>45360</td><td> 2 4 ⋅ 3 4 ⋅ 5 ⋅ 7 {\displaystyle 2^{4}\cdot 3^{4}\cdot 5\cdot 7} </td><td>4,4,1,1</td><td>10</td><td>100</td><td> 6 3 ⋅ 210 {\displaystyle 6^{3}\cdot 210} </td></tr><tr><td>27</td><td>50400</td><td> 2 5 ⋅ 3 2 ⋅ 5 2 ⋅ 7 {\displaystyle 2^{5}\cdot 3^{2}\cdot 5^{2}\cdot 7} </td><td>5,2,2,1</td><td>10</td><td>108</td><td> 2 3 ⋅ 30 ⋅ 210 {\displaystyle 2^{3}\cdot 30\cdot 210} </td></tr><tr><td>28</td><td>55440*</td><td> 2 4 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{4}\cdot 3^{2}\cdot 5\cdot 7\cdot 11} </td><td>4,2,1,1,1</td><td>9</td><td>120</td><td> 2 2 ⋅ 6 ⋅ 2310 {\displaystyle 2^{2}\cdot 6\cdot 2310} </td></tr><tr><td>29</td><td>83160</td><td> 2 3 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{3}\cdot 3^{3}\cdot 5\cdot 7\cdot 11} </td><td>3,3,1,1,1</td><td>9</td><td>128</td><td> 6 2 ⋅ 2310 {\displaystyle 6^{2}\cdot 2310} </td></tr><tr><td>30</td><td>110880</td><td> 2 5 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{5}\cdot 3^{2}\cdot 5\cdot 7\cdot 11} </td><td>5,2,1,1,1</td><td>10</td><td>144</td><td> 2 3 ⋅ 6 ⋅ 2310 {\displaystyle 2^{3}\cdot 6\cdot 2310} </td></tr><tr><td>31</td><td>166320</td><td> 2 4 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{4}\cdot 3^{3}\cdot 5\cdot 7\cdot 11} </td><td>4,3,1,1,1</td><td>10</td><td>160</td><td> 2 ⋅ 6 2 ⋅ 2310 {\displaystyle 2\cdot 6^{2}\cdot 2310} </td></tr><tr><td>32</td><td>221760</td><td> 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{6}\cdot 3^{2}\cdot 5\cdot 7\cdot 11} </td><td>6,2,1,1,1</td><td>11</td><td>168</td><td> 2 4 ⋅ 6 ⋅ 2310 {\displaystyle 2^{4}\cdot 6\cdot 2310} </td></tr><tr><td>33</td><td>277200</td><td> 2 4 ⋅ 3 2 ⋅ 5 2 ⋅ 7 ⋅ 11 {\displaystyle 2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7\cdot 11} </td><td>4,2,2,1,1</td><td>10</td><td>180</td><td> 2 2 ⋅ 30 ⋅ 2310 {\displaystyle 2^{2}\cdot 30\cdot 2310} </td></tr><tr><td>34</td><td>332640</td><td> 2 5 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{5}\cdot 3^{3}\cdot 5\cdot 7\cdot 11} </td><td>5,3,1,1,1</td><td>11</td><td>192</td><td> 2 2 ⋅ 6 2 ⋅ 2310 {\displaystyle 2^{2}\cdot 6^{2}\cdot 2310} </td></tr><tr><td>35</td><td>498960</td><td> 2 4 ⋅ 3 4 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{4}\cdot 3^{4}\cdot 5\cdot 7\cdot 11} </td><td>4,4,1,1,1</td><td>11</td><td>200</td><td> 6 3 ⋅ 2310 {\displaystyle 6^{3}\cdot 2310} </td></tr><tr><td>36</td><td>554400</td><td> 2 5 ⋅ 3 2 ⋅ 5 2 ⋅ 7 ⋅ 11 {\displaystyle 2^{5}\cdot 3^{2}\cdot 5^{2}\cdot 7\cdot 11} </td><td>5,2,2,1,1</td><td>11</td><td>216</td><td> 2 3 ⋅ 30 ⋅ 2310 {\displaystyle 2^{3}\cdot 30\cdot 2310} </td></tr><tr><td>37</td><td>665280</td><td> 2 6 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 {\displaystyle 2^{6}\cdot 3^{3}\cdot 5\cdot 7\cdot 11} </td><td>6,3,1,1,1</td><td>12</td><td>224</td><td> 2 3 ⋅ 6 2 ⋅ 2310 {\displaystyle 2^{3}\cdot 6^{2}\cdot 2310} </td></tr><tr><td>38</td><td>720720*</td><td> 2 4 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 {\displaystyle 2^{4}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13} </td><td>4,2,1,1,1,1</td><td>10</td><td>240</td><td> 2 2 ⋅ 6 ⋅ 30030 {\displaystyle 2^{2}\cdot 6\cdot 30030} </td></tr><tr><td>39</td><td>1081080</td><td> 2 3 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 {\displaystyle 2^{3}\cdot 3^{3}\cdot 5\cdot 7\cdot 11\cdot 13} </td><td>3,3,1,1,1,1</td><td>10</td><td>256</td><td> 6 2 ⋅ 30030 {\displaystyle 6^{2}\cdot 30030} </td></tr><tr><td>40</td><td>1441440*</td><td> 2 5 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 {\displaystyle 2^{5}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13} </td><td>5,2,1,1,1,1</td><td>11</td><td>288</td><td> 2 3 ⋅ 6 ⋅ 30030 {\displaystyle 2^{3}\cdot 6\cdot 30030} </td></tr><tr><td>41</td><td>2162160</td><td> 2 4 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 {\displaystyle 2^{4}\cdot 3^{3}\cdot 5\cdot 7\cdot 11\cdot 13} </td><td>4,3,1,1,1,1</td><td>11</td><td>320</td><td> 2 ⋅ 6 2 ⋅ 30030 {\displaystyle 2\cdot 6^{2}\cdot 30030} </td></tr></tbody></table> <p>The divisors of the first 20 highly composite numbers are shown below. </p> <table><tbody><tr><th><i>n</i></th><th><i>d</i>(<i>n</i>)</th><th>Divisors of <i>n</i></th></tr><tr><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>2</td><td>1, 2</td></tr><tr><td>4</td><td>3</td><td>1, 2, 4</td></tr><tr><td>6</td><td>4</td><td>1, 2, 3, 6</td></tr><tr><td>12</td><td>6</td><td>1, 2, 3, 4, 6, 12</td></tr><tr><td>24</td><td>8</td><td>1, 2, 3, 4, 6, 8, 12, 24</td></tr><tr><td>36</td><td>9</td><td>1, 2, 3, 4, 6, 9, 12, 18, 36</td></tr><tr><td>48</td><td>10</td><td>1, 2, 3, 4, 6, 8, 12, 16, 24, 48</td></tr><tr><td>60</td><td>12</td><td>1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60</td></tr><tr><td>120</td><td>16</td><td>1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120</td></tr><tr><td>180</td><td>18</td><td>1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180</td></tr><tr><td>240</td><td>20</td><td>1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240</td></tr><tr><td>360</td><td>24</td><td>1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360</td></tr><tr><td>720</td><td>30</td><td>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</td></tr><tr><td>840</td><td>32</td><td>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</td></tr><tr><td>1260</td><td>36</td><td>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</td></tr><tr><td>1680</td><td>40</td><td>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</td></tr><tr><td>2520</td><td>48</td><td>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</td></tr><tr><td>5040</td><td>60</td><td>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</td></tr><tr><td>7560</td><td>64</td><td>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</td></tr></tbody></table> <p>The table below shows all 72 divisors of 10080 by writing it as a product of two numbers in 36 different ways. </p> <table><tbody><tr><td colspan="6">The highly composite number: 10080 10080 = (2 × 2 × 2 × 2 × 2) × (3 × 3) × 5 × 7</td></tr><tr><td>1×10080</td><td>2 × 5040</td><td>3 × 3360</td><td>4 × 2520</td><td>5 × 2016</td><td>6 × 1680</td></tr><tr><td>7× 1440</td><td>8 × 1260</td><td>9 × 1120</td><td>10 × 1008</td><td>12 × 840</td><td>14 × 720</td></tr><tr><td>15× 672</td><td>16 × 630</td><td>18 × 560</td><td>20 × 504</td><td>21 × 480</td><td>24 × 420</td></tr><tr><td>28× 360</td><td>30 × 336</td><td>32 × 315</td><td>35 × 288</td><td>36 × 280</td><td>40 × 252</td></tr><tr><td>42× 240</td><td>45 × 224</td><td>48 × 210</td><td>56 × 180</td><td>60 × 168</td><td>63 × 160</td></tr><tr><td>70× 144</td><td>72 × 140</td><td>80 × 126</td><td>84 × 120</td><td>90 × 112</td><td>96 × 105</td></tr><tr><td colspan="6"><i>Note: </i> Numbers in bold are themselves highly composite numbers. Only the twentieth highly composite number 7560 (= 3 × 2520) is absent.10080 is a so-called <a href="/facts/Smooth_number/pC6DVLJW">7-smooth number</a> <i>(sequence A002473 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>)</i>.</td></tr></tbody></table> <p>The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes: </p> 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 ⋯ a 229 , {\displaystyle 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},} <p>where a n {\displaystyle a_{n}} is the n {\displaystyle n} th successive prime number, and all omitted terms (<i>a</i>22 to <i>a</i>228) are factors with exponent equal to one (i.e. the number is 2 14 × 3 9 × 5 6 × ⋯ × 1451 {\displaystyle 2^{14}\times 3^{9}\times 5^{6}\times \cdots \times 1451} ). More concisely, it is the product of seven distinct primorials: </p> b 0 5 b 1 3 b 2 2 b 4 b 7 b 18 b 229 , {\displaystyle b_{0}^{5}b_{1}^{3}b_{2}^{2}b_{4}b_{7}b_{18}b_{229},} <p>where b n {\displaystyle b_{n}} is the <a href="/facts/Primorial/NqlQBLDp">primorial</a> a 0 a 1 ⋯ a n {\displaystyle a_{0}a_{1}\cdots a_{n}} .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="prime-factorization">Prime factorization</h2> <p>Roughly speaking, for a number to be highly composite it has to have <a href="/facts/Prime_factors/L20FLkgn">prime factors</a> as small as possible, but not too many of the same. By the <a href="/facts/Fundamental_theorem_of_arithmetic/EzHUwlXX">fundamental theorem of arithmetic</a>, every positive integer <i>n</i> has a unique prime factorization: </p> n = p 1 c 1 × p 2 c 2 × ⋯ × p k c k {\displaystyle n=p_{1}^{c_{1}}\times p_{2}^{c_{2}}\times \cdots \times p_{k}^{c_{k}}} <p>where p 1 < p 2 < ⋯ < p k {\displaystyle p_{1}<p_{2}<\cdots <p_{k}} are prime, and the exponents c i {\displaystyle c_{i}} are positive integers. </p><p>Any factor of n must have the same or lesser multiplicity in each prime: </p> p 1 d 1 × p 2 d 2 × ⋯ × p k d k , 0 ≤ d i ≤ c i , 0 < i ≤ k {\displaystyle p_{1}^{d_{1}}\times p_{2}^{d_{2}}\times \cdots \times p_{k}^{d_{k}},0\leq d_{i}\leq c_{i},0<i\leq k} <p>So the number of divisors of <i>n</i> is: </p> d ( n ) = ( c 1 + 1 ) × ( c 2 + 1 ) × ⋯ × ( c k + 1 ) . {\displaystyle d(n)=(c_{1}+1)\times (c_{2}+1)\times \cdots \times (c_{k}+1).} <p>Hence, for a highly composite number <i>n</i>, </p> <ul><li>the <i>k</i> given prime numbers <i>p</i><i>i</i> must be precisely the first <i>k</i> prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than <i>n</i> with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors);</li> <li>the sequence of exponents must be non-increasing, that is c 1 ≥ c 2 ≥ ⋯ ≥ c k {\displaystyle c_{1}\geq c_{2}\geq \cdots \geq c_{k}} ; otherwise, by exchanging two exponents we would again get a smaller number than <i>n</i> with the same number of divisors (for instance 18 = 21 × 32 may be replaced with 12 = 22 × 31; both have six divisors).</li></ul> <p>Also, except in two special cases <i>n</i> = 4 and <i>n</i> = 36, the last exponent <i>c</i><i>k</i> must equal 1. It means that 1, 4, and 36 are the only square highly composite numbers. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of <a href="/facts/Primorials/NqlQBLDp">primorials</a> or, alternatively, the smallest number for its <a href="/facts/Prime_signature/kmRpC7F8">prime signature</a>. </p><p>Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example, 96 = 25 × 3 satisfies the above conditions and has 12 divisors but is not highly composite since there is a smaller number (60) which has the same number of divisors. </p> <h2 id="asymptotic-growth-and-density">Asymptotic growth and density</h2> <p>If <i>Q</i>(<i>x</i>) denotes the number of highly composite numbers less than or equal to <i>x</i>, then there are two constants <i>a</i> and <i>b</i>, both greater than 1, such that </p> ( log x ) a ≤ Q ( x ) ≤ ( log x ) b . {\displaystyle (\log x)^{a}\leq Q(x)\leq (\log x)^{b}\,.} <p>The first part of the inequality was proved by <a href="/facts/Paul_Erd%25C5%2591s/63LoUTe1">Paul Erdős</a> in 1944 and the second part by <a href="/facts/Jean-Louis_Nicolas/3XJz1XLX">Jean-Louis Nicolas</a> in 1988. We have </p> 1.13862 < lim inf log Q ( x ) log log x ≤ 1.44 {\displaystyle 1.13862<\liminf {\frac {\log Q(x)}{\log \log x}}\leq 1.44\ } <p>and </p> lim sup x → ∞ log Q ( x ) log log x ≤ 1.71 . {\displaystyle \limsup _{x\,\to \,\infty }{\frac {\log Q(x)}{\log \log x}}\leq 1.71\ .} <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> <h2 id="related-sequences">Related sequences</h2> <p>Highly composite numbers greater than 6 are also <a href="/facts/Abundant_numbers/K54Q9owk">abundant numbers</a>. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also <a href="/facts/Harshad_numbers/Xa1eUGg5">Harshad numbers</a> in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800. </p><p>10 of the first 38 highly composite numbers are <a href="/facts/Superior_highly_composite_numbers/NzD4z9QY">superior highly composite numbers</a>. The sequence of highly composite numbers (sequence A002182 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>) is a subset of the sequence of smallest numbers <i>k</i> with exactly <i>n</i> divisors (sequence A005179 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>). </p><p>Highly composite numbers whose number of divisors is also a highly composite number are </p> 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 (sequence A189394 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>). <p>It is extremely likely that this sequence is complete. </p><p>A positive integer <i>n</i> is a largely composite number if <i>d</i>(<i>n</i>) ≥ <i>d</i>(<i>m</i>) for all <i>m</i> ≤ <i>n</i>. The counting function <i>Q</i><i>L</i>(<i>x</i>) of largely composite numbers satisfies </p> ( log x ) c ≤ log Q L ( x ) ≤ ( log x ) d {\displaystyle (\log x)^{c}\leq \log Q_{L}(x)\leq (\log x)^{d}\ } <p>for positive <i>c</i> and <i>d</i> with 0.2 ≤ c ≤ d ≤ 0.5 {\displaystyle 0.2\leq c\leq d\leq 0.5} .<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>Because the prime factorization of a highly composite number uses all of the first <i>k</i> primes, every highly composite number must be a <a href="/facts/Practical_number/SSRVtFPp">practical number</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Due to their ease of use in calculations involving <a href="/facts/Fraction_(mathematics)/ghnQE9B3">fractions</a>, many of these numbers are used in <a href="/facts/Historical_weights_and_measures/xIsWmFWW">traditional systems of measurement</a> and engineering designs. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Superior_highly_composite_number/NzD4z9QY">Superior highly composite number</a></li> <li><a href="/facts/Highly_totient_number/Vtlfrl2F">Highly totient number</a></li> <li><a href="/facts/Table_of_divisors/3RXGmBtn">Table of divisors</a></li> <li><a href="/facts/Euler%2527s_totient_function/mIQwXYzj">Euler's totient function</a></li> <li><a href="/facts/Round_number/IvFP6hlr">Round number</a></li> <li><a href="/facts/Smooth_number/pC6DVLJW">Smooth number</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). <i>Handbook of number theory I</i>. Dordrecht: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. pp. 45–46. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4020-4215-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1151.11300">1151.11300</a>.</li> <li><a href="/facts/Paul_Erd%25C5%2591s/63LoUTe1">Erdös, P.</a> (1944). <a href="https://www.renyi.hu/~p_erdos/1944-04.pdf">"On highly composite numbers"</a> (PDF). <i><a href="/facts/Journal_of_the_London_Mathematical_Society/jdcf3vhX">Journal of the London Mathematical Society</a></i>. Second Series. 19 (75_Part_3): 130–133. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fjlms%2F19.75_part_3.130">10.1112/jlms/19.75_part_3.130</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0013381">0013381</a>.</li> <li><a href="/facts/Leonidas_Alaoglu/kSMacT2N">Alaoglu, L.</a>; <a href="/facts/Paul_Erd%25C5%2591s/63LoUTe1">Erdös, P.</a> (1944). <a href="https://www.renyi.hu/~p_erdos/1944-03.pdf">"On highly composite and similar numbers"</a> (PDF). <i><a href="/facts/Transactions_of_the_American_Mathematical_Society/JmKtyCwK">Transactions of the American Mathematical Society</a></i>. 56 (3): 448–469. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1990319">10.2307/1990319</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1990319">1990319</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0011087">0011087</a>.</li> <li><a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Ramanujan, Srinivasa</a> (1997). <a href="http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf">"Highly composite numbers"</a> (PDF). <i>Ramanujan Journal</i>. 1 (2): 119–153. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1009764017495">10.1023/A:1009764017495</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1606180">1606180</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:115619659">115619659</a>. Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/HighlyCompositeNumber.html">"Highly Composite Number"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="https://web.archive.org/web/19980707133810/http://www.math.princeton.edu/~kkedlaya/math/hcn-algorithm.tex">Algorithm for computing Highly Composite Numbers</a></li> <li><a href="https://web.archive.org/web/19980707133953/http://www.math.princeton.edu/~kkedlaya/math/hcn10000.txt.gz">First 10000 Highly Composite Numbers as factors</a></li> <li><a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Achim Flammenkamp, First 779674 HCN with sigma, tau, factors</a></li> <li><a href="http://www.javascripter.net/math/calculators/highlycompositenumbers.htm">Online Highly Composite Numbers Calculator</a></li> <li><a href="https://www.youtube.com/watch?v=2JM2oImb9Qg">5040 and other Anti-Prime Numbers - Dr. James Grime</a> by Dr. James Grime for <a href="/facts/Numberphile/pqqG2edI">Numberphile</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Ramanujan, S. (1915). "Highly composite numbers" (PDF). Proc. London Math. Soc. Series 2. 14: 347–409. doi:10.1112/plms/s2_14.1.347. JFM 45.1248.01. <a href="/wiki/Srinivasa_Ramanujan" target="_blank">/wiki/Srinivasa_Ramanujan</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Kahane, Jean-Pierre (February 2015), "Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'oeuvre", Notices of the American Mathematical Society, 62 (2): 136–140. Kahane cites Plato's Laws, 771c. <a href="/wiki/Jean-Pierre_Kahane" target="_blank">/wiki/Jean-Pierre_Kahane</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Vardoulakis, Antonis; Pugh, Clive (September 2008), "Plato's hidden theorem on the distribution of primes", The Mathematical Intelligencer, 30 (3): 61–63, doi:10.1007/BF02985381. <a href="https://link.springer.com/article/10.1007/BF02985381" target="_blank">https://link.springer.com/article/10.1007/BF02985381</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Flammenkamp, Achim, Highly Composite Numbers. <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html" target="_blank">http://wwwhomes.uni-bielefeld.de/achim/highly.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Sándor et al. (2006) p. 45 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Sándor et al. (2006) p. 46 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Nicolas, Jean-Louis (1979). "Répartition des nombres largement composés". Acta Arith. (in French). 34 (4): 379–390. doi:10.4064/aa-34-4-379-390. Zbl 0368.10032. <a href="/wiki/Jean-Louis_Nicolas" target="_blank">/wiki/Jean-Louis_Nicolas</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Srinivasan, A. K. (1948), "Practical numbers" (PDF), Current Science, 17: 179–180, MR 0027799. <a href="http://www.ias.ac.in/jarch/currsci/17/179.pdf" target="_blank">http://www.ias.ac.in/jarch/currsci/17/179.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>