HIGHLY COMPOSITE NUMBER

__NOTOC__
A 'highly composite number' (HCN) is a positive integer which has more divisors than any positive integer below it. (There is a second use of the term; see the section below.)
The first twenty-one highly composite numbers are listed in the table at right.

$n$	number of divisors of $n$
1	1
2	2
4	3
6	4
12	6
24	8
36	9
48	10
60	12
120	16
180	18
240	20
360	24
720	30
840	32
1260	36
1680	40
2520	48
5040	60
7560	64
10080	72

The sequence of highly composite numbers is a subset of the sequence of smallest numbers ''k'' with exactly ''n'' divisors .
There are an infinite number of highly composite numbers. To prove this fact, suppose that ''n'' is an arbitrary highly composite number. Then 2''n'' has more divisors than ''n'' (2''n itself is a divisor and so are all the divisors of ''n'') and so some number larger than ''n'' (and not larger than 2''n'') must be highly composite as well.
Roughly speaking, for a number to be a highly composite it has to have prime factors as small as possible, but not too many of the same. If we decompose a number ''n'' in prime factors like this:
:$n\; =\; p\_1^\{c\_1\}\; imes\; p\_2^\{c\_2\}\; imes\; cdots\; imes\; p\_k^\{c\_k\}$ (1)
where are prime, and the exponents $c\_i$ are positive integers, then the number of divisors of ''n'' is exactly
:$(c\_1\; +\; 1)\; imes\; (c\_2\; +\; 1)\; imes\; cdots\; imes\; (c\_k\; +\; 1)$. (2)
Hence, for ''n'' to be a highly composite number,

★ 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 4 divisors);

★ the sequence of exponents must be non-increasing, that is $c\_1\; geq\; c\_2\; geq\; cdots\; geq\; c\_k$; 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 6 divisors).
Also, except in two special cases ''n'' = 4 and ''n'' = 36, the last exponent ''c''_''k'' must equal 1. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials.
Highly composite numbers higher than 6 are also abundant numbers. One need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. All highly composite numbers are also Harshad numbers in base 10.
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 vulgar fractions.
If ''Q''(''x'') denotes the number of highly composite numbers which are less than or equal to ''x'', then there exist two constants ''a'' and ''b'', both bigger than 1, so that
:(ln ''x'')^''a'' ≤ ''Q''(''x'') ≤ (ln ''x'')^''b''.
with the first part of the inequality proved by Paul Erdős in 1944 and the second part by J.-L. Nicholas in 1988.

Contents

Example

Second definition

Related

See also

External links

Example

' The highly composite number :  10080.' 10080  =  (2 × 2 × 2 × 2 × 2)  ×  (3 × 3)  ×  5  ×  7 'By (2) above, 10080 has exactly seventy-two divisors.'
'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 'bold'ed numbers are themselves 'highly composite numbers'. Only the twentieth highly composite number 7560 (=3×2520) is absent.

10080 is a so-called 7-smooth number, ''.

The 15,000th Highly Composite Number is the product: $$
a_{0}^{15} a_{1}^{10} a_{2}^{6} a_{3}^{5} a_{4}^{3} 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}...a_{229} a_{230}, where $a\_n$ is the sequence of successive prime numbers, and all omitted terms are factors with exponent equal to one (i.e. $2^\{15\}\; imes\; 3^\{10\}\; imes\; 5^6\; imes\; ...\; imes\; 1451\; imes\; 1453$).

Second definition

There is a second use of the term ''highly composite number'', defined as a number with all prime divisors ≤ 7. The first few terms are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, and 27 . These are also called '7-smooth numbers'; see Smooth number for a generalization and applications.

Related

One interesting characteristic of the HCNs is that quite often there is a prime number immediately adjacent. Number 120 is the first without an adjacent prime number. In addition, a conjecture says the distance from a HCN to the nearest prime when >1 will itself be a prime number (the distance must always be odd due to involving an odd and even number).

See also

★ Superior highly composite number

★ Table of divisors

External links

★ MathWorld: Highly Composite Number

★ Earth360: Versatile Numbers: Self-Organization, Emergence, and Economics

★ Algorithm for computing Highly Composite Numbers

★ First 10000 Highly Composite Numbers

★ First 1200 HCN with sigma,tau,factors