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Properties based on the sum-of-divisors function s(n), including perfect and multiperfect numbers, aliquot sequences, amicable pairs and sociable chains.

Mersenne@

• Aliquot Sequences with initial term up to 10,000, by Christophe Clavier.
• Aliquot Sequences Statistics by Wolfgang Creyaufmueller, including many tables and links to factoring sites and software.
• 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.
• 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.
• On a Generalization of Perfect Numbers J. L. Pe introduces perfect numbers relative to an arithmetical function f. Under this scheme, the usual perfect numbers are just one among many species of "f-perfect numbers". Several open problems and examples of perfect number sets are given, as well as a few f-amicable pairs.
• 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.
• Perfect Numbers Harry J. Smith's pages of the mathematics behind perfect numbers, aliquot sequences and Mersenne numbers. Includes program listings and e-mail archives.
• Perfect, Amicable and Sociable Numbers Definitions, links and references of various applications of the sum of divisors function s(n).
• Tables of Aliquot Cycles Data for various kinds up to 10^12, including unitary, infinitary, exponential, augmented and reduced varieties.
• 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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