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| {{about|numbers having many divisors|numbers factorized only to powers of 2, 3, 5 and 7 (also named 7-smooth numbers)|Smooth number}}
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| __NOTOC__
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| A '''highly composite number''' ('''HCN''') is a [[Positive number|positive]] [[integer]] with more [[divisor]]s than any smaller positive integer.
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| The initial or smallest twenty-six highly composite numbers are listed in the table at right.
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| {| class="wikitable" style="text-align:center" align="right"
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| ! Order
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| ! <math>n</math>
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| ! prime number<br> factorization
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| ! number of <br>divisors of <math>n</math>
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| |-
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| | 1
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| | [[1 (number)|1]]
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| | 1
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| |-
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| | 2
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| | [[2 (number)|2]]
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| | <math>2</math>
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| | 2
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| |-
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| | 3
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| | [[4 (number)|4]]
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| | <math>2^2</math>
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| | 3
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| |-
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| | 4
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| | [[6 (number)|6]]
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| | <math>2\cdot 3</math>
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| | 4
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| |-
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| | 5
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| | [[12 (number)|12]]
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| | <math>2^2\cdot 3</math>
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| | 6
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| |-
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| | 6
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| | [[24 (number)|24]]
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| | <math>2^3\cdot 3</math>
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| | 8
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| |-
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| | 7
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| | [[36 (number)|36]]
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| | <math>2^2\cdot 3^2</math>
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| | 9
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| |-
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| | 8
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| | [[48 (number)|48]]
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| | <math>2^4\cdot 3</math>
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| | 10
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| |-
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| | 9
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| | [[60 (number)|60]]
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| | <math>2^2\cdot 3\cdot 5</math>
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| | 12
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| |-
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| | 10
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| | [[120 (number)|120]]
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| | <math>2^3\cdot 3\cdot 5</math>
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| | 16
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| |-
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| | 11
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| | [[180 (number)|180]]
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| | <math>2^2\cdot 3^2\cdot 5</math>
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| | 18
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| |-
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| | 12
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| | [[240 (number)|240]]
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| | <math>2^4\cdot 3\cdot 5</math>
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| | 20
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| |-
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| | 13
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| | [[360 (number)|360]]
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| | <math>2^3\cdot 3^2\cdot 5</math>
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| | 24
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| |-
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| | 14
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| | [[720 (number)|720]]
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| |<math>2^4\cdot 3^2\cdot 5</math>
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| | 30
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| |-
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| | 15
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| | 840
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| | <math>2^3\cdot 3\cdot 5\cdot 7</math>
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| | 32
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| |-
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| | 16
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| | 1,260
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| | <math>2^2\cdot 3^2\cdot 5\cdot 7</math>
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| | 36
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| |-
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| | 17
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| | 1,680
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| | <math>2^4\cdot 3\cdot 5\cdot 7</math>
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| | 40
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| |-
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| | 18
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| | [[2520 (number)|2,520]]
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| | <math>2^3\cdot 3^2\cdot 5\cdot 7</math>
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| | 48
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| |-
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| | 19
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| | [[5040 (number)|5,040]]
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| | <math>2^4\cdot 3^2\cdot 5\cdot 7</math>
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| | 60
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| |-
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| | 20
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| | 7,560
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| | <math>2^3\cdot 3^3\cdot 5\cdot 7</math>
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| | 64
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| |-
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| | 21
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| | 10,080
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| | <math>2^5\cdot 3^2\cdot 5\cdot 7</math>
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| | 72
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| |-
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| | 22
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| | 15,120
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| | <math>2^4\cdot 3^3\cdot 5\cdot 7</math>
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| | 80
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| |-
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| | 23
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| | 20,160
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| | <math>2^6\cdot 3^2\cdot 5\cdot 7</math>
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| | 84
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| |-
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| | 24
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| | 25,200
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| | <math>2^4\cdot 3^2\cdot 5^2\cdot 7</math>
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| | 90
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| |-
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| | 25
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| | 27,720
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| | <math>2^3\cdot 3^2\cdot 5\cdot 7\cdot 11</math>
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| | 96
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| |-
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| | 26
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| | 45,360
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| | <math>2^4\cdot 3^4\cdot 5\cdot 7</math>
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| | 100
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| |}
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| The sequence of highly composite numbers {{OEIS|id=A002182}} is a subset of the sequence of smallest numbers ''k'' with exactly ''n'' divisors {{OEIS|id=A005179}}.
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| There are an [[Infinity|infinite]] number of highly composite numbers. To [[mathematical proof|prove]] this fact, suppose that ''n'' is an arbitrary highly composite number. Then 2''n'' has more divisors than ''n'' (2''n'' itself is a divisor and so are all the divisors of ''n'') and so some number larger than ''n'' (and not larger than 2''n'') must be highly composite as well.
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| Roughly speaking, for a number to be highly composite it has to have [[prime factor]]s as small as possible, but not too many of the same. If we decompose a number ''n'' in prime factors like this:
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| :<math>n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}\qquad (1)</math>
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| where <math>p_1 < p_2 < \cdots < p_k</math> are prime, and the exponents <math>c_i</math> are positive integers, then the number of divisors of ''n'' is exactly
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| :<math>(c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1).\qquad (2)</math>
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| Hence, for ''n'' to be a highly composite number,
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| * the ''k'' given prime numbers ''p''<sub>''i''</sub> must be precisely the first ''k'' 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 ''n'' with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors);
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| * the sequence of exponents must be non-increasing, that is <math>c_1 \geq c_2 \geq \cdots \geq c_k</math>; otherwise, by exchanging two exponents we would again get a smaller number than ''n'' with the same number of divisors (for instance 18 = 2<sup>1</sup> × 3<sup>2</sup> may be replaced with 12 = 2<sup>2</sup> × 3<sup>1</sup>; both have six divisors).
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| Also, except in two special cases ''n'' = 4 and ''n'' = 36, the last exponent ''c''<sub>''k''</sub> must equal 1. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of [[primorials]]. Because the prime factorization of a highly composite number uses all of the first ''k'' primes, every highly composite number must be a [[practical number]].<ref>{{citation
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| | last = Srinivasan | first = A. K.
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| | title = Practical numbers
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| | journal = [[Current Science]]
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| | volume = 17
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| | year = 1948
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| | pages = 179–180
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| | id = {{MathSciNet | id = 0027799}}
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| | url = http://www.ias.ac.in/jarch/currsci/17/179.pdf}}.</ref>
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| Highly composite numbers higher than 6 are also [[abundant number]]s. One need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also [[Harshad number]]s in base 10. The first HCN that is not a Harshad number is 245,044,800, which has a digit sum of 27, but 27 does not divide evenly into 245,044,800.
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| Many of these numbers are used in [[historical weights and measures|traditional systems of measurement]], and tend to be used in engineering designs, due to their ease of use in calculations involving [[fraction (mathematics)|fraction]]s.
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| If ''Q''(''x'') denotes the number of highly composite numbers less than or equal to ''x'', then there are two constants ''a'' and ''b'', both greater than 1, such that
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| :<math>\ln(x)^a \le Q(x) \le \ln(x)^b \, .</math>
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| The first part of the inequality was proved by [[Paul Erdős]] in 1944 and the second part by Jean-Louis Nicolas in 1988. We have<ref name=HBI45>Sándor et al (2006) p.45</ref>
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| :<math>1.13862 < \liminf \frac{\log Q(x)}{\log\log x} \le 1.44 \ </math>
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| and
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| :<math>\limsup \frac{\log Q(x)}{\log\log x} \le 1.71 \ .</math>
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| ==Examples==
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| {| class="wikitable" style="text-align:center;table-layout:fixed;"
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| |colspan="6"| <big>'''The highly composite number: 10,080'''</big> <br /> 10,080 = (2 × 2 × 2 × 2 × 2) × (3 × 3) × 5 × 7 <br /> By (2) above, 10,080 has exactly seventy-two divisors.
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| |- style="color:#000000;background:#ffffff;"
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| |style="line-height:1.4" height=64|'''1'''<br />×<br />'''10,080
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| |style="line-height:1.4"| '''2''' <br /> × <br /> ''' 5,040
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| |style="line-height:1.4"| 3 <br /> × <br /> 3,360
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| |style="line-height:1.4"| '''4''' <br /> × <br /> ''' 2,520
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| |style="line-height:1.4"| 5 <br /> × <br /> 2,016
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| |style="line-height:1.4"| '''6''' <br /> × <br /> ''' 1,680
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| |- style="color:#000000;background:#ffffff;"
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| |style="line-height:1.4" height=64|7<br />× <br /> 1,440
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| |style="line-height:1.4"| 8 <br /> × <br /> ''' 1,260
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| |style="line-height:1.4"| 9 <br /> × <br /> 1,120
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| |style="line-height:1.4"| 10 <br /> × <br /> 1,008
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| |style="line-height:1.4"| '''12''' <br /> × <br /> ''' 840
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| |style="line-height:1.4"| 14 <br /> × <br /> ''' 720
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| |- style="color:#000000;background:#ffffff;"
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| |style="line-height:1.4" height=64|15<br />×<br /> 672
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| |style="line-height:1.4"| 16 <br /> × <br /> 630
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| |style="line-height:1.4"| 18 <br /> × <br /> 560
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| |style="line-height:1.4"| 20 <br /> × <br /> 504
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| |style="line-height:1.4"| 21 <br /> × <br /> 480
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| |style="line-height:1.4"| '''24''' <br /> × <br /> 420
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| |- style="color:#000000;background:#ffffff;"
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| |style="line-height:1.4" height=64|28<br />×<br /> ''' 360
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| |style="line-height:1.4"| 30 <br /> × <br /> 336
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| |style="line-height:1.4"| 32 <br /> × <br /> 315
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| |style="line-height:1.4"| 35 <br /> × <br /> 288
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| |style="line-height:1.4"| '''36''' <br /> × <br /> 280
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| |style="line-height:1.4"| 40 <br /> × <br /> 252
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| |- style="color:#000000;background:#ffffff;"
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| |style="line-height:1.4" height=64|42<br />×<br /> ''' 240
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| |style="line-height:1.4"| 45 <br /> × <br /> 224
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| |style="line-height:1.4"| '''48''' <br /> × <br /> 210
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| |style="line-height:1.4"| 56 <br /> × <br /> ''' 180
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| |style="line-height:1.4"| '''60''' <br /> × <br /> 168
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| |style="line-height:1.4"| 63 <br /> × <br /> 160
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| |- style="color:#000000;background:#ffffff;"
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| |style="line-height:1.4" height=64|70<br />×<br /> 144
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| |style="line-height:1.4"| 72 <br /> × <br /> 140
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| |style="line-height:1.4"| 80 <br /> × <br /> 126
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| |style="line-height:1.4"| 84 <br /> × <br /> ''' 120
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| |style="line-height:1.4"| 90 <br /> × <br /> 112
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| |style="line-height:1.4"| 96 <br /> × <br /> 105
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| |colspan="6"|'''''Note: ''''' Numbers in '''bold''' are themselves '''highly composite numbers'''. <br /> Only the twentieth highly composite number 7560 (= 3 × 2520) is absent.<br />10080 is a so-called [[smooth number|7-smooth number]] ''{{OEIS|id=A002473}}''.
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| |}
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| The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:
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| : <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>
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| where <math>a_n</math> is the sequence of successive prime numbers, 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>). <ref>{{citation
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| | last = Flammenkamp | first = Achim
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| | title = Highly Composite Numbers
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| | url = http://wwwhomes.uni-bielefeld.de/achim/highly.html}}.</ref>
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| ==Prime factor subsets==
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| For any highly composite number, if one takes any subset of prime factors for that number and their exponents, the resulting number will have more divisors than any smaller number that uses the same prime factors. For example for the highly composite number 720 which is 2<sup>4</sup> × 3<sup>2</sup> × 5 we can be sure that
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| * 144 which is 2<sup>4</sup> × 3<sup>2</sup> has more divisors than any smaller number that has only the prime factors 2 and 3
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| * 80 which is 2<sup>4</sup> × 5 has more divisors than any smaller number that has only the prime factors 2 and 5
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| * 45 which is 3<sup>2</sup> × 5 has more divisors than any smaller number that has only the prime factors 3 and 5
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| If this were untrue for any particular highly composite number and subset of prime factors, we could exchange that subset of prime factors and exponents for the smaller number using the same prime factors and get a smaller number with at least as many divisors.
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| This property is useful for finding highly composite numbers.
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| ==See also==
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| * [[Abundant number]]
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| * [[Highly totient number]]
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| * [[Superior highly composite number]]
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| * [[Table of divisors]]
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| * [[Euler's totient function]]
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| ==References==
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| {{reflist}}
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| * {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | pages=45–46}}
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| == External links ==
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| * {{MathWorld |urlname=HighlyCompositeNumber |title=Highly Composite Number}}
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| * [http://web.archive.org/web/19980707133810/www.math.princeton.edu/~kkedlaya/math/hcn-algorithm.tex Algorithm for computing Highly Composite Numbers]
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| * [http://web.archive.org/web/19980707133953/www.math.princeton.edu/~kkedlaya/math/hcn10000.txt.gz First 10000 Highly Composite Numbers]
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| * [http://wwwhomes.uni-bielefeld.de/achim/highly.html Achim Flammenkamp, First 779674 HCN with sigma,tau,factors]
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| * [http://www.javascripter.net/math/calculators/highlycompositenumbers.htm Online Highly Composite Numbers Calculator]
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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| [[Category:Conjectures]]
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