# 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.[1][2]

## Examples

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* ${\displaystyle 2}$ 1 1 2 ${\displaystyle 2}$
3 4 ${\displaystyle 2^{2}}$ 2 2 3 ${\displaystyle 2^{2}}$
4 6* ${\displaystyle 2\cdot 3}$ 1,1 2 4 ${\displaystyle 6}$
5 12* ${\displaystyle 2^{2}\cdot 3}$ 2,1 3 6 ${\displaystyle 2\cdot 6}$
6 24 ${\displaystyle 2^{3}\cdot 3}$ 3,1 4 8 ${\displaystyle 2^{2}\cdot 6}$
7 36 ${\displaystyle 2^{2}\cdot 3^{2}}$ 2,2 4 9 ${\displaystyle 6^{2}}$
8 48 ${\displaystyle 2^{4}\cdot 3}$ 4,1 5 10 ${\displaystyle 2^{3}\cdot 6}$
9 60* ${\displaystyle 2^{2}\cdot 3\cdot 5}$ 2,1,1 4 12 ${\displaystyle 2\cdot 30}$
10 120* ${\displaystyle 2^{3}\cdot 3\cdot 5}$ 3,1,1 5 16 ${\displaystyle 2^{2}\cdot 30}$
11 180 ${\displaystyle 2^{2}\cdot 3^{2}\cdot 5}$ 2,2,1 5 18 ${\displaystyle 6\cdot 30}$
12 240 ${\displaystyle 2^{4}\cdot 3\cdot 5}$ 4,1,1 6 20 ${\displaystyle 2^{3}\cdot 30}$
13 360* ${\displaystyle 2^{3}\cdot 3^{2}\cdot 5}$ 3,2,1 6 24 ${\displaystyle 2\cdot 6\cdot 30}$
14 720 ${\displaystyle 2^{4}\cdot 3^{2}\cdot 5}$ 4,2,1 7 30 ${\displaystyle 2^{2}\cdot 6\cdot 30}$
15 840 ${\displaystyle 2^{3}\cdot 3\cdot 5\cdot 7}$ 3,1,1,1 6 32 ${\displaystyle 2^{2}\cdot 210}$
16 1260 ${\displaystyle 2^{2}\cdot 3^{2}\cdot 5\cdot 7}$ 2,2,1,1 6 36 ${\displaystyle 6\cdot 210}$
17 1680 ${\displaystyle 2^{4}\cdot 3\cdot 5\cdot 7}$ 4,1,1,1 7 40 ${\displaystyle 2^{3}\cdot 210}$
18 2520* ${\displaystyle 2^{3}\cdot 3^{2}\cdot 5\cdot 7}$ 3,2,1,1 7 48 ${\displaystyle 2\cdot 6\cdot 210}$
19 5040* ${\displaystyle 2^{4}\cdot 3^{2}\cdot 5\cdot 7}$ 4,2,1,1 8 60 ${\displaystyle 2^{2}\cdot 6\cdot 210}$
20 7560 ${\displaystyle 2^{3}\cdot 3^{3}\cdot 5\cdot 7}$ 3,3,1,1 8 64 ${\displaystyle 6^{2}\cdot 210}$
21 10080 ${\displaystyle 2^{5}\cdot 3^{2}\cdot 5\cdot 7}$ 5,2,1,1 9 72 ${\displaystyle 2^{3}\cdot 6\cdot 210}$
22 15120 ${\displaystyle 2^{4}\cdot 3^{3}\cdot 5\cdot 7}$ 4,3,1,1 9 80 ${\displaystyle 2\cdot 6^{2}\cdot 210}$
23 20160 ${\displaystyle 2^{6}\cdot 3^{2}\cdot 5\cdot 7}$ 6,2,1,1 10 84 ${\displaystyle 2^{4}\cdot 6\cdot 210}$
24 25200 ${\displaystyle 2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7}$ 4,2,2,1 9 90 ${\displaystyle 2^{2}\cdot 30\cdot 210}$
25 27720 ${\displaystyle 2^{3}\cdot 3^{2}\cdot 5\cdot 7\cdot 11}$ 3,2,1,1,1 8 96 ${\displaystyle 2\cdot 6\cdot 2310}$
26 45360 ${\displaystyle 2^{4}\cdot 3^{4}\cdot 5\cdot 7}$ 4,4,1,1 10 100 ${\displaystyle 6^{3}\cdot 210}$
27 50400 ${\displaystyle 2^{5}\cdot 3^{2}\cdot 5^{2}\cdot 7}$ 5,2,2,1 10 108 ${\displaystyle 2^{3}\cdot 30\cdot 210}$
28 55440* ${\displaystyle 2^{4}\cdot 3^{2}\cdot 5\cdot 7\cdot 11}$ 4,2,1,1,1 9 120 ${\displaystyle 2^{2}\cdot 6\cdot 2310}$
29 83160 ${\displaystyle 2^{3}\cdot 3^{3}\cdot 5\cdot 7\cdot 11}$ 3,3,1,1,1 9 128 ${\displaystyle 6^{2}\cdot 2310}$
30 110880 ${\displaystyle 2^{5}\cdot 3^{2}\cdot 5\cdot 7\cdot 11}$ 5,2,1,1,1 10 144 ${\displaystyle 2^{3}\cdot 6\cdot 2310}$
31 166320 ${\displaystyle 2^{4}\cdot 3^{3}\cdot 5\cdot 7\cdot 11}$ 4,3,1,1,1 10 160 ${\displaystyle 2\cdot 6^{2}\cdot 2310}$
32 221760 ${\displaystyle 2^{6}\cdot 3^{2}\cdot 5\cdot 7\cdot 11}$ 6,2,1,1,1 11 168 ${\displaystyle 2^{4}\cdot 6\cdot 2310}$
33 277200 ${\displaystyle 2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7\cdot 11}$ 4,2,2,1,1 10 180 ${\displaystyle 2^{2}\cdot 30\cdot 2310}$
34 332640 ${\displaystyle 2^{5}\cdot 3^{3}\cdot 5\cdot 7\cdot 11}$ 5,3,1,1,1 11 192 ${\displaystyle 2^{2}\cdot 6^{2}\cdot 2310}$
35 498960 ${\displaystyle 2^{4}\cdot 3^{4}\cdot 5\cdot 7\cdot 11}$ 4,4,1,1,1 11 200 ${\displaystyle 6^{3}\cdot 2310}$
36 554400 ${\displaystyle 2^{5}\cdot 3^{2}\cdot 5^{2}\cdot 7\cdot 11}$ 5,2,2,1,1 11 216 ${\displaystyle 2^{3}\cdot 30\cdot 2310}$
37 665280 ${\displaystyle 2^{6}\cdot 3^{3}\cdot 5\cdot 7\cdot 11}$ 6,3,1,1,1 12 224 ${\displaystyle 2^{3}\cdot 6^{2}\cdot 2310}$
38 720720* ${\displaystyle 2^{4}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13}$ 4,2,1,1,1,1 10 240 ${\displaystyle 2^{2}\cdot 6\cdot 30030}$

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).

## 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.[4][5]

## Notes

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.