Highly abundant number

Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944).

For example, but there is a smaller number with larger sum of divisors, so 9!

Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant.

7200 is the largest powerful number that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once.

Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors.