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:
“Subject the material world to the higher ends by understanding it in all its relations to daily life and action.”
—Ellen Henrietta Swallow Richards (1842–1911)
“I would rather have as my patron a host of anonymous citizens digging into their own pockets for the price of a book or a magazine than a small body of enlightened and responsible men administering public funds. I would rather chance my personal vision of truth striking home here and there in the chaos of publication that exists than attempt to filter it through a few sets of official, honorably public-spirited scruples.”
—John Updike (b. 1932)
“The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.”
—Claude Lévi-Strauss (b. 1908)