Highly composite number

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. (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.

Failed to parse (Missing texvc executable; please see math/README to configure.): n	number of divisors of Failed to parse (Missing texvc executable; please see math/README to configure.): 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 (sequence A002182 in OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in OEIS).

There are an infinite number of highly composite numbers. To prove this fact, suppose that n is an arbitrary highly composite number. Then 2n has more divisors than n (2n itself is a divisor and so are all the divisors of n) and so some number larger than n (and not larger than 2n) 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}

(1)

where Failed to parse (Missing texvc executable; please see math/README to configure.): p_1 < p_2 < \cdots < p_k

are prime, and the exponents Failed to parse (Missing texvc executable; please see math/README to configure.): c_i
are positive integers, then the number of divisors of n is exactly

Failed to parse (Missing texvc executable; please see math/README to configure.): (c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1)

. (2)

Hence, for n to be a highly composite number,

the k given prime numbers Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

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 bolded 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, (sequence A002473 in OEIS).

The 15,000th Highly Composite Number is the product: Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): a_n

is the sequence of successive prime numbers, and all omitted terms are factors with exponent equal to one (i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): 2^{15}  \times 3^{10}  \times 5^6  \times ... \times 1451 \times 1453

).^{[citation needed]}

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 (sequence A002473 in OEIS). These are also called 7-smooth numbers; see Smooth number for a generalization and applications.

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).