Colossally abundant number
Definition and Properties
In number theory, a colossally abundant number (CAN) is a natural number that, in a certain technical sense, has many divisors. The concept of colossally abundant numbers is a refinement of the concept of highly abundant numbers, which are numbers for which the sum of their divisors (including the number itself) is greater than the sum for any smaller number. Colossally abundant numbers were first introduced by Srinivasa Ramanujan in 1915.
The formal definition of a colossally abundant number is as follows: a natural number n is said to be colossally abundant if there is a positive exponent ε such that for all k > 0, the inequality
σ(n)/n^ε ≥ σ(k)/k^ε
holds, where σ denotes the divisor function, which is the sum of the divisors of a number.
Properties of Colossally Abundant Numbers
Colossally abundant numbers have several interesting properties that set them apart from other types of numbers.
One of the most important properties of colossally abundant numbers is that they are highly composite, meaning they have more divisors than any smaller positive integer. This is a direct consequence of the definition of colossally abundant numbers, as the abundance of a number is directly related to the number of its divisors.
Another important property of colossally abundant numbers is that they are always even. This is because any odd number has fewer divisors than the even number that immediately follows it, so the even number will always be more abundant.
Examples of Colossally Abundant Numbers
The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, and 5040.
The number 2 is the smallest colossally abundant number. It has only two divisors, 1 and 2, but the sum of these divisors is 3, which is greater than the sum of the divisors of 1.
The number 6 is the next colossally abundant number. It has four divisors: 1, 2, 3, and 6. The sum of these divisors is 12, which is greater than the sum of the divisors of any smaller number.
Relationship with Other Types of Numbers
Colossally abundant numbers are closely related to several other types of numbers in number theory.
As mentioned earlier, all colossally abundant numbers are also highly abundant numbers. However, the converse is not true: not all highly abundant numbers are colossally abundant.
Colossally abundant numbers are also closely related to superior highly composite numbers. A superior highly composite number is a number that has more divisors than any smaller number and also more divisors per unit than any smaller number. Every colossally abundant number is a superior highly composite number, but not every superior highly composite number is a colossally abundant number.
Applications and Significance
Colossally abundant numbers have several applications in number theory and other areas of mathematics.
One of the most significant applications of colossally abundant numbers is in the study of the Riemann zeta function. The Riemann zeta function has a deep connection with the distribution of prime numbers, and colossally abundant numbers play a key role in understanding this connection.
Colossally abundant numbers also have applications in the study of factorization and divisibility, two fundamental concepts in number theory.