Relations With Other Sets of Numbers
Although the first eight factorials are highly abundant, 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,
so 9! is not highly abundant.
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. This question was answered affirmatively by Nicolas (1969).
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers is abundant.
Read more about this topic: Highly Abundant Number
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