Highly composite number

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.. Kahane cites Plato's Laws, 771c.

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 HCNn prime factorization primeexponents numberof primefactors  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 7204,2,1 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.

 The highly composite number: 10080  10080 = (2 × 2 × 2 × 2 × 2)  ×  (3 × 3)  ×  5  ×  71×10080  2  ×   5040     3     ×      3360  4  ×   2520     5     ×      2016  6  ×   16807×      1440     8     ×   1260     9     ×      1120    10     ×      1008 12  ×    840    14     ×    72015×      672    16     ×       630    18     ×       560    20     ×       504    21     ×       480 24  ×       42028×   360    30     ×       336    32     ×       315    35     ×       288 36  ×       280    40     ×       25242×   240    45     ×       224 48  ×       210    56     ×    180 60  ×       168    63     ×       16070×      144    72     ×       140    80     ×       126    84     ×    120    90     ×       112    96     ×       105Note:  The numbers in bold are also highly composite numbers. 10080 is often referred to as a 7-smooth number .
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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.Sándor et al. (2006) p. 46

 Related pages 
 Abundant number
 Euler's totient function
 Harshad number

Notes

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

Category:Integer sequences