Highly composite number
If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N. For example, 6 is highly composite because d(6)=4 and d(n)=1,2,2,3,2 for n=1,2,3,4,5 respectively.
Ramanujan wrote a paper on highly composite numbers in 1915.
[2] Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040.
[3] The first 41 highly composite numbers are listed in the table below (sequence A002182 in the OEIS).
Asterisks indicate superior highly composite numbers.
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.
[4] 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
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, Also, except in two special cases n = 4 and n = 36, the last exponent ck 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 or, alternatively, the smallest number for its prime signature.
Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite.
If Q(x) 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 that The first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988.
One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact.
A positive integer n is a largely composite number if d(n) ≥ d(m) for all m ≤ n. The counting function QL(x) of largely composite numbers satisfies for positive c and d with
[8] Due to their ease of use in calculations involving fractions, many of these numbers are used in traditional systems of measurement and engineering designs.
