In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to some positive power of itself than any other number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. The first 10 superior highly composite numbers and their factorization are listed. For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k smaller than n we have and for all natural numbers k larger than n we have where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan (1915). For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12. 120 is another superior highly composite number because it has the highest ratio of divisors to itself raised to the .4 power. The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither set, however, is a subset of the other. Properties All superior highly composite numbers are highly composite. This is easy to prove: if there is some number k that has the same number of divisors as n but is less than n itself (i.e. , but ), then for all positive ε, so if a number "n" is not highly composite, it cannot be superior highly composite. An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers. Let for any prime number p and positive real x. Then is a superior highly composite number. Note that the product need not be computed indefinitely, because if then , so the product to calculate can be terminated once . Also note that in the definition of , is analogous to in the implicit definition of a superior highly composite number. Moreover, for each superior highly composite number exists a half-open interval such that . This representation implies that there exist an infinite sequence of such that for the n-th