In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural 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). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N. They also proved that 7200 is the largest powerful highly abundant number, and therefore the largest highly abundant number with odd sum of divisors.
Read more about Highly Abundant Number: Formal Definition and Examples, Relations With Other Sets of Numbers
Famous quotes containing the words highly, abundant and/or number:
“Disdain and scorn ride sparkling in her eyes,
Misprizing what they look on, and her wit
Values itself so highly that to her
All matter else seems weak.”
—William Shakespeare (1564–1616)
“It was a flourishing tropic he required
For his refreshment, an abundant zone,
Prickly and obdurate, dense, harmonious,
Yet with a harmony not rarefied
Nor fined for the inhibited instruments
Of over-civil stops.”
—Wallace Stevens (1879–1955)
“[The] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)