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# Highly composite number

A highly composite number is a positive integer which has more divisors than any positive integer below it. The first twenty highly composite numbers are

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240 , 360, 720, 840, 1260, 1680, 2520, 5040, 7560 and 10080.

, with 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64 and 72 positive divisors, respectively (sequence A002183 in OEIS). The sequence of highly composite numbers 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 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:

$n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}$

where $p_1 < p_2 < \cdots < p_k$ are prime, and the exponents ci are positive integers, then the number of divisors of n is exactly

$(c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1)$.

Hence, for n to be a highly composite number,

• the k given prime numbers pi 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 faulty exponents we would again get a smaller number than n with the same number of divisors (for instance 18=21x32 may be replaced with 12=22x31, both have 6 divisors).

Also, except in two special cases n = 4 and n = 36, the last exponent ck must equate 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.

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.

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)aQ(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.