Superior highly composite number

In mathematics, a superior highly composite number is a natural number which has more divisors than any other number scaled relative to some power of the number itself. 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).

The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence in the OEIS) 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.

All superior highly composite numbers are 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

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