1. Examples

2. Prime factorization

3. Asymptotic growth and density

4. Related sequences

5. See also

6. Notes

7. References

8. External links

A highly composite number, also known as an anti-prime,^[1] 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.^[2]

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 {{OEIS|id=A002182}}. The number of divisors is given in the column labeled d(n).

Order	HCN n	prime factorization	prime exponents	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 table below shows all the divisors of one of these numbers.

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

where 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 ). More concisely, it is the product of seven distinct primorials:

where is the primorial .

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:

where are prime, and the exponents are positive integers.

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

So the number of divisors of n is:

Hence, for a highly composite number n,

• the k given prime numbers p_i 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 ; 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 c_k 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

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^[4]

and

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 {{OEIS|id=A002182}} is a subset of the sequence of smallest numbers k with exactly n divisors {{OEIS|id=A005179}}.

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 {{OEIS|id=A189394}}. 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 m ≤ n. The counting function Q_L(x) of largely composite numbers satisfies

for positive c,d with .^[5][6]

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.^[7] 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.

See also

• Superior highly composite number
• Highly totient number
• Table of divisors
• Euler's totient function
• Round number
• Smooth number

Notes

References

• {{cite journal | last1=Ramanujan | first1=S. | author1-link=Srinivasa Ramanujan | title=Highly composite numbers | jfm=45.1248.01 | doi=10.1112/plms/s2_14.1.347 | journal=Proc. London Math. Soc. |series=Series 2 | volume=14 | pages=347–409 | year=1915}} (online)
• {{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}}
• {{cite journal

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External links

• {{MathWorld |urlname=HighlyCompositeNumber |title=Highly Composite Number}}
• [https://web.archive.org/web/19980707133810/http://www.math.princeton.edu/~kkedlaya/math/hcn-algorithm.tex Algorithm for computing Highly Composite Numbers]
• [https://web.archive.org/web/19980707133953/http://www.math.princeton.edu/~kkedlaya/math/hcn10000.txt.gz First 10000 Highly Composite Numbers as factors]
• Achim Flammenkamp, First 779674 HCN with sigma,tau,factors
• Online Highly Composite Numbers Calculator

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