Superior highly composite number
Encyclopedia
In mathematics
, a superior highly composite number is a certain kind of natural number
with the following properties. A natural number n is called superior highly composite if there is an ε > 0 such that for all natural numbers k ≥ 1,
where d(n), the divisor function
, denotes the number of divisors of n. The first few superior highly composite numbers are 2, 6, 12
, 60
, 120
, 360
, 2520, 5040
, 55440, 720720, 1441440, 4324320, 21621600, 367567200... .
; it can also be shown that there exist prime number
s π1, π2, ... such that the n-th superior highly composite number sn can be written as
The first few πn are 2, 3, 2, 5, 2, 3, 7, ... . In other words, the ratio between two successive superior highly composite numbers is equal to a prime number.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a superior highly composite number is a certain kind of natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
with the following properties. A natural number n is called superior highly composite if there is an ε > 0 such that for all natural numbers k ≥ 1,
where d(n), the divisor function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
, denotes the number of divisors of n. The first few superior highly composite numbers are 2, 6, 12
12 (number)
12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
, 60
60 (number)
60 is the natural number following 59 and preceding 61. Being three times twenty, 60 is called "three score" in some older literature.-In mathematics:...
, 120
120 (number)
120 is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120. The number 100, now known commonly as "one hundred" was then known as "the small hundred". It is also known as...
, 360
360 (number)
360 or three sixty is the natural number following 359 and preceding 361.-In mathematics:*Divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360. *360 makes a highly composite number...
, 2520, 5040
5040 (number)
5040 is a factorial , a highly composite number, a superior highly composite number, a colossally abundant number, and the number of permutations of 4 items out of 10 choices .-Philosophy:...
, 55440, 720720, 1441440, 4324320, 21621600, 367567200... .
Properties
All superior highly composite numbers are highly compositeHighly composite number
A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.The initial or smallest twenty-one highly composite numbers are listed in the table at right....
; it can also be shown that there exist prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s π1, π2, ... such that the n-th superior highly composite number sn can be written as
The first few πn are 2, 3, 2, 5, 2, 3, 7, ... . In other words, the ratio between two successive superior highly composite numbers is equal to a prime number.