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A highly composite number (HCN) is a Positive Integer which has more Divisor s 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.

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 p_1 < p_2 < \cdots < p_k 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 Number s. 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 Number s 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 Fraction s.

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 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



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