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 smaller 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 divisor than any numbers less than it.[1][2]

Examples[change | change source]

The first 38 highly composite numbers are listed in the table below (sequence A002182 in the OEIS). 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
d(n) 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 d(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  ×  7
1
×
10080
2
×
5040
3
×
3360
4
×
2520
5
×
2016
6
×
1680
7
×
1440
8
×
1260
9
×
1120
10
×
1008
12
×
840
14
×
720
15
×
672
16
×
630
18
×
560
20
×
504
21
×
480
24
×
420
28
×
360
30
×
336
32
×
315
35
×
288
36
×
280
40
×
252
42
×
240
45
×
224
48
×
210
56
×
180
60
×
168
63
×
160
70
×
144
72
×
140
80
×
126
84
×
120
90
×
112
96
×
105
Note:  The numbers in bold are also highly composite numbers. 10080 is often referred to as a 7-smooth number (sequence A002473 in the OEIS).

[3]

Similar sequences[change | change source]

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.[4][5]

Related pages[change | change source]

Notes[change | change source]

  1. 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.
  2. 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.
  3. Flammenkamp, Achim, Highly Composite Numbers.
  4. Sándor et al. (2006) p. 46
  5. 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.

References[change | change source]

Other websites[change | change source]