superior highly composite number on:
[Wikipedia]
[Google]
[Amazon]
In mathematics, a superior highly composite number is a 
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
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
which has the highest ratio of its number of divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s 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
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includi ...
, 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,