# Highly composite number

A highly composite number, sometimes called an antiprime number, is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it.[1]

Demonstration, with Cuisenaire rods, of the first four: 1, 2, 4, 6

The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer.

The name can be somewhat misleading, as two highly composite numbers (1 and 2) are not actually composite numbers.

## Examples

The initial or smallest 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). Asterisks indicate 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 divisors 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:  Numbers in bold are themselves highly composite numbers. Only the twentieth highly composite number 7560 (= 3 × 2520) is absent.10080 is a so-called 7-smooth number (sequence A002473 in the OEIS).

The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:

${\displaystyle a_{0}^{14}a_{1}^{9}a_{2}^{6}a_{3}^{4}a_{4}^{4}a_{5}^{3}a_{6}^{3}a_{7}^{3}a_{8}^{2}a_{9}^{2}a_{10}^{2}a_{11}^{2}a_{12}^{2}a_{13}^{2}a_{14}^{2}a_{15}^{2}a_{16}^{2}a_{17}^{2}a_{18}^{2}a_{19}a_{20}a_{21}\cdots a_{229},}$

where ${\displaystyle a_{n}}$  is the sequence of successive prime numbers, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is ${\displaystyle 2^{14}\times 3^{9}\times 5^{6}\times \cdots \times 1451}$ ). More concisely, it is the product of seven distinct primorials:

${\displaystyle b_{0}^{5}b_{1}^{3}b_{2}^{2}b_{4}b_{7}b_{18}b_{229},}$

where ${\displaystyle b_{n}}$  is the primorial ${\displaystyle a_{0}a_{1}\cdots a_{n}}$ . [2]

Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are labelled in bold and superior highly composite numbers are starred. In the SVG file, hover over a bar to see its statistics.

## Prime factorization

Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization:

${\displaystyle n=p_{1}^{c_{1}}\times p_{2}^{c_{2}}\times \cdots \times p_{k}^{c_{k}}\qquad (1)}$

where ${\displaystyle p_{1}  are prime, and the exponents ${\displaystyle c_{i}}$  are positive integers.

Any factor of n must have the same or lesser multiplicity in each prime:

${\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

So the number of divisors of n is:

${\displaystyle d(n)=(c_{1}+1)\times (c_{2}+1)\times \cdots \times (c_{k}+1).\qquad (2)}$

Hence, for a highly composite number n,

• the k given prime numbers pi 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);
• the sequence of exponents must be non-increasing, that is ${\displaystyle c_{1}\geq c_{2}\geq \cdots \geq c_{k}}$ ; otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors (for instance 18 = 21 × 32 may be replaced with 12 = 22 × 31; both have six divisors).

Also, except in two special cases n = 4 and n = 36, the last exponent ck 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 primorials.

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.

## Asymptotic growth and density

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

${\displaystyle (\log x)^{a}\leq Q(x)\leq (\log x)^{b}\,.}$

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[3]

${\displaystyle 1.13862<\liminf {\frac {\log Q(x)}{\log \log x}}\leq 1.44\ }$

and

${\displaystyle \limsup {\frac {\log Q(x)}{\log \log x}}\leq 1.71\ .}$

## Related sequences

Highly composite numbers higher than 6 are also abundant numbers. 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 Harshad numbers 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.

10 of the first 38 highly composite numbers are superior highly composite numbers. The sequence of highly composite numbers (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS).

Highly composite numbers whose number of divisors is also a highly composite number are for n = 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 (sequence A189394 in the OEIS). It is extremely likely that this sequence is complete.

A positive integer n is a largely composite number if d(n) ≥ d(m) for all mn. The counting function QL(x) of largely composite numbers satisfies

${\displaystyle (\log x)^{c}\leq \log Q_{L}(x)\leq (\log x)^{d}\ }$

for positive c,d with ${\displaystyle 0.2\leq c\leq d\leq 0.5}$ .[4][5]

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.[6] Many of these numbers are used in traditional systems of measurement, and tend to be used in engineering designs, due to their ease of use in calculations involving fractions.