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
Famous quotes containing the words relations, sets and/or numbers:
“Words are but symbols for the relations of things to one another and to us; nowhere do they touch upon absolute truth.”
—Friedrich Nietzsche (1844–1900)
“To me this world is all one continued vision of fancy or imagination, and I feel flattered when I am told so. What is it sets Homer, Virgil and Milton in so high a rank of art? Why is Bible more entertaining and instructive than any other book? Is it not because they are addressed to the imagination, which is spiritual sensation, and but mediately to the understanding or reason?”
—William Blake (1757–1827)
“And when all bodies meet
In Lethe to be drowned,
Then only numbers sweet
With endless life are crowned.”
—Robert Herrick (1591–1674)