In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n.
Six (6) is the first perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 .
These first four perfect numbers were the only ones known to the ancient Greeks.
More on [ Perfect number ]

Aliquot - Sequences with initial term up to 10,000, by Christophe Clavier.
Aliquot Sequences - Juan L. Varona's work on aliquot sequences examines sequences with start values up to 10000, and traces them to large composite values.
Aliquot Sequences - Results and software from Paul Zimmermann, addressing the Catalan Conjecture and working on the difficult Lehmer Five sequences.
Aliquot Sequences - Aliquot sequences arise in iterating the sum-of-divisors function s. Bosma has calculated tables up to start value 50,000 and 80 digits.
Aliquot Sequences - Statistics by Wolfgang Creyaufmueller, including many tables and links to factoring sites and software.
Meta Description: [ To proof Catalans conjecture needs many aliquot sequences.
The programmes (download) factorize up to 120 digits. ]
Amicable Numbers - Observations on amicable pairs and their distribution, including Harshad and Happy Amicable pairs.
Cubes of Perfection - From Ivars Peterson's MathTrek column in MAA Online. Curious relationships satisfied by perfect numbers.
Multiply Perfect Numbers - Up-to-date information on multiperfect numbers and searchable database maintained by Achim Flammenkamp.
Odd Perfect Numbers - Missing Values - Status of various factorisations that would extend the known area of non-existence of odd perfect numbers, by Kevin Hare.
Perfect Number - from MathWorld - Definitions and references.
Perfect Numbers - A detailed history of the quest for perfect numbers, from Euclid to their present-day Mersenne discoveries.
Tables of Aliquot Cycles - Data for various kinds up to 10^12, including unitary, infinitary, exponential, augmented and reduced varieties.
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The Perfect Number Journey - A Mathematics Enrichment Workshop introducing perfect numbers. Lessons and exercises extend over several pages. Aimed at interested middle and high school students.
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