Highly Abundant Number

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:

    As far as the arts and the sciences are concerned, the German mind appreciates most highly that which it does not understand of the latter, and that which it does not enjoy of the former.
    Franz Grillparzer (1791–1872)

    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)

    Cultivated labor drives out brute labor. An infinite number of shrewd men, in infinite years, have arrived at certain best and shortest ways of doing, and this accumulated skill in arts, cultures, harvestings, curings, manufactures, navigations, exchanges, constitutes the worth of our world to-day.
    Ralph Waldo Emerson (1803–1882)