Highly composite number
A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan. 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.
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 are not actually composite numbers.
Examples
The initial or smallest 38 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled d. Asterisks indicate superior highly composite numbers.| Order | HCN n | prime factorization | prime exponents | number of prime factors | d | 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 divisors of the first 15 highly composite numbers are shown below.
| n | d | 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 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 are factors with exponent equal to one. More concisely, it is the product of seven distinct primorials:
where is the primorial.
File:Highly_composite_numbers.svg|thumb|250px|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 , 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: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 pi must be precisely the first k prime numbers ; 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 ;
- 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.
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 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 thatThe first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988. We have
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 is a subset of the sequence of smallest numbers k with exactly n divisors.
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. It is extremely likely that this sequence is complete.
A positive integer n is a largely composite number if d ≥ d for all m ≤ n. The counting function QL of largely composite numbers satisfies
for positive c,d with.
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. 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.