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Perfect Numbers
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Statistics by Wolfgang Creyaufmueller, including many tables and links to factoring sites and software.
http://home.t-online.de/home/Wol...gang.Creyaufmueller/aliquote.htm
Juan L. Varona's work on aliquot sequences examines sequences with start values up to 10000, and traces them to large composite values.
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Aliquot sequences W page about aliquot sequences. First of all, sorry for the English in this page; I am sure it will have some spelling and grammar mistakes. Also, note that the notation for some mathematical formulas follows the one of TeX....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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Mathematics Enrichment Workshop: The Perfect Number Journey Lessons on number patterns and properties of numbers by Heng O.K. 6, 28, 496 What are perfect numbers? Mathematicians and nonmathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are...A detailed history of the quest for perfect numbers, from Euclid to their present-day Mersenne discoveries.
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Perfect numbers It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity. It is quite likely, although not certain, that the Egyptians would have come...Results and software from Paul Zimmermann, addressing the Catalan Conjecture and working on the difficult "Lehmer Five" sequences.
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Aliquot Sequences The aliquot sequence starting from n is defined as follows: let &sigma(n) be the sum of divisors of n, then one simply computes f(n sigma(n)-n, and one iterates. For example, if we start from 12, whose divisors are 1, 2,...From Ivars Peterson's MathTrek column in MAA Online. Curious relationships satisfied by perfect numbers.
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Ivars Peterson's MathTrek Cubes of Perfection Playing with integers can lead to all sorts of little surprises. A whole number that is equal to the sum of all its possible divisors including 1 but not the number itself is known as a perfect number (see A...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.
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On a Generalization of Perfect Numbers This paper presents a notion of perfect numbers relative to arithmetical functions: an arithmetical function f produces a set of f-perfect numbers. Two among the many examples considered are small “perturbations” of the normal definition; late in these two sequences,...Observations on amicable pairs and their distribution, including Harshad and Happy Amicable pairs.
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Amicable Numbers by Shyam Sunder Gupta What is a friend replied that a friend is one who is the other I such as 220 and 284. The numbers 220 and 284 form the smallest pair of amicable numbers (also known as friendly numbers) known to Pythagoras....Up-to-date information on multiperfect numbers and searchable database maintained by Achim Flammenkamp.
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Multiply Perfect Numbers A multiply perfect number is called proper if its abundancy is 2. For example consider the divisors of the number 120: 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120=o(120)=o(2^3*3*5)=o(2^3)*o(3)*o(5 1+2+4+8 1+3 1+5)=15*4*6=360=3*120. Hence 120 is a 3-fold perfect number. Status Abundancy Count When last number was discovered...Data for various kinds up to 10^12, including unitary, infinitary, exponential, augmented and reduced varieties.
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Tables of Aliquot Cycles Click on the number of known cycles to view the corresponding lists. Type Amicable Numbers Sociable Numbersof order four Sociable Numbersof other orders Perfect Numbers KnownCycles LastUpdate KnownCycles LastUpdate KnownCycles LastUpdate KnownCycles LastUpdate Ordinary 10306909 04-Oct-2005 137 10-Oct-2005 7 18-Oct-1997...Sequences with initial term up to 10,000, by Christophe Clavier.
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Aliquot sequences (with leading terme 10,000) There are other pages related to aliquot sequences, and maintained by: Wieb Bosma Chris K. Caldwell Wolfgang Creyaufmueller Sander Hoogendoorn Jim Howell David Moews Clifford Stern Juan Luis Varona Eric W. Weisstein Paul Zimmermann Note that the seven sequences 276,...Definitions and references.
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Perfect Number From MathWorld Perfect numbers are positive integers such that (1) where is the restricted divisor function (i.e the sum of proper divisors of or equivalently (2) where is the divisor function (i.e the sum of divisors of including itself For example, the...Status of various factorisations that would extend the known area of non-existence of odd perfect numbers, by Kevin Hare.
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This page out of date If you are interested in the current status, please contact me. kghare at math dot uwaterloo dot ca....Aliquot sequences arise in iterating the sum-of-divisors function s. Bosma has calculated tables up to start value 50,000 and 80 digits.
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Aliquot sequences arise in iterating the sum-of-divisors function s, which assigns to a positive integer the sum of its proper divisors (i.e excluding the number itself An aliquot sequence thus starts with a positive integer n, followed by s(n then...Harry J. Smith's pages of the mathematics behind perfect numbers, aliquot sequences and Mersenne numbers. Includes program listings and e-mail archives.
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Perfect Numbers from Harry J. Smith This page accessed times since June 7, 1997. Page created by: hjsmithh@sbcglobal.net Changes last made on Friday, 13-May-05 21:06:17 PDT...Statistics by Wolfgang Creyaufmueller, including many tables and links to factoring sites and software.
http://www.aliquot.de/aliquote.htm