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 with, relations, sets and/or numbers:
“If one could be friendly with women, what a pleasure—the relationship so secret and private compared with relations with men. Why not write about it truthfully?”
—Virginia Woolf (1882–1941)
“I only desire sincere relations with the worthiest of my acquaintance, that they may give me an opportunity once in a year to speak the truth.”
—Henry David Thoreau (1817–1862)
“Nothing sets a person up more than having something turn out just the way it’s supposed to be, like falling into a Swiss snowdrift and seeing a big dog come up with a little cask of brandy round its neck.”
—Claud Cockburn (1904–1981)
“I’m not even thinking straight any more. Numbers buzz in my head like wasps.”
—Kurt Neumann (1906–1958)