Highly abundant number

Highly abundant numbers are a type of natural number. Any natural number n is called highly abundant when a certain equation is true.

In this equation, m is every integer less than n. σ is the sum of every positive divisor of that number.
Example
An example would be to use the number 5. σ of 5 is 5 + 1 = 6. Every σ less than 5, however, is 4 + 2 + 1 = 7. 7 is greater than 6. This makes 5 not a highly abundant number.
History
Highly abundant numbers were first learned about by Subbayya Sivasankaranarayana Pillai in 1943. Work on the subject was done by Paul Erdős and Leonidas Alaoglu in 1944. Alaoglu and Erdős discovered every highly abundant number up to 104. They also showed that the number of highly abundant numbers less than N is proportional to log2 N.

The first few highly abundant numbers are

 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... .

There are only two odd highly abundant numbers. They are 1 and 3.

 Similarities with other numbers 
The first eight factorials are highly abundant. However, not all factorials are highly abundant. For example,

 σ(9!) = σ(362880) = 1481040,

but there is a smaller number with larger sum of divisors,

 σ(360360) = 1572480.

This makes 9! is not highly abundant.

Alaoglu and Erdős discovered that all superabundant numbers are also highly abundant. There are, however, an infinite number of highly abundant numbers that are not superabundant numbers. This was proven by Jean-Louis Nicolas in 1969.

7200 is the largest powerful number that is also highly abundant. This is because every highly abundant number that is larger has a prime factor that divides them only once., pp. 464–466.

 Notes 

 References 





Category:Integer sequences
Category:Divisor function