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	Perfect number (Site not responding. Last check: 2007-10-13)
		A perfect number is an integer which is the sum of its proper positive divisors (factors), not including the number itself.
		Nowadays, prime numbers generated by the formula are known as Mersenne primes, after the seventeenth-century monk, Marin Mersenne, who studied number theory and perfect numbers.
		Numbers where the sum is less than the number itself are called deficient, and where it is greater, abundant; these terms, together with perfect itself, come from Greek numerology.
	bopedia.com /en/wikipedia/p/pe/perfect_number.html (588 words)

	Informat.io on Perfect Number
		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.
		Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant.
		By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.
	www.informat.io /?title=perfect-number (1199 words)

	PlanetMath: quasiperfect number
		is the number of distinct prime factors function).
		A quasiperfect number would thus overshoot the mark for being a perfect number by a margin of just 1.
		This is version 3 of quasiperfect number, born on 2006-07-20, modified 2006-10-02.
	planetmath.org /encyclopedia/QuasiperfectNumber.html (95 words)

	Search ScienceWorld
		A palindromic number is a number (in some base b) that is the same when written forwards or backwards, i.e., of the form a_1a_2...a_2a_1.
		Quasiperfect numbers are therefore the sum of their nontrivial divisors.
		No quasiperfect numbers are known, although if any exist, they must be greater than 10^(35) and have seven or more distinct prime factors.
	scienceworld.wolfram.com /search/index.cgi?num=&q=Number&start=160 (497 words)

	UCSD Math Club - Fun & Games
		A natural number n is called a perfect number if the sum of its proper divisors add to n.
		Now a natural number n is called a quasiperfect number if the sum of its proper divisors is n+1.
		Prove, however, that if there exists a quasiperfect number n, then it must be an odd perfect square (that is, n must satisfy n = (2m + 1)^2 for some integer m).
	math.ucsd.edu /~mathclub/games/brainteaser-archive/quasi.html (599 words)

	PlanetMath: abundant number
		Given a pair of amicable numbers, the lesser of the two is abundant.
		positive multiple of an abundant number is abundant
		This is version 2 of abundant number, born on 2006-04-25, modified 2006-04-26.
	planetmath.org /encyclopedia/AbundantNumber.html (108 words)

	Straight Dope Staff Report: What's the story on perfect numbers? (Site not responding. Last check: 2007-10-13)
		Perfect numbers are a holdover from the days of the Pythagoreans, when mathematicians were mystics as much as anything else and put a lot more stock in coincidence.
		Quasiperfect numbers have a divisor sum of 2n+1; almost perfect numbers have a divisor sum of 2n-1.
		For example, sublime numbers have a perfect number of divisors, and the sum of their divisors is itself perfect.
	www.straightdope.com /mailbag/mperfectnumbers.html (683 words)

	Perfect number - ExampleProblems.com
		N is of the form 12j + 1 or 36j + 9 (Jacques Touchard).
		Numbers where the sum is less than twice the number itself are called deficient, and where it is greater than twice the number, abundant.
		A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.
	www.exampleproblems.com /wiki/index.php/Perfect_number (846 words)

	Perfect Number -- from MathWorld
		Perfect numbers were deemed to have important numerological properties by the ancients, and were extensively studied by the Greeks, including Euclid.
		Perfect numbers are also intimately connected with a class of numbers known as Mersenne primes, which are prime numbers of the form
		It is known that all even perfect numbers (except 6) end in 16, 28, 36, 56, 76, or 96 (Lucas 1891) and have digital root 1.
	users.skynet.be /fa956617/math/topics/PerfectNumber.html (710 words)

	math lessons - Category:Number theory
		Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians.
		More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers.
		Number theory may be subdivided into several fields according to the methods used and the questions investigated.
	www.mathdaily.com /lessons/Category:Number_theory (102 words)

	I Can't Think of a Clever Title: Wiki Wednesday
		Its divisors are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60.
		Abundant numbers are part of a branch of mathematics called number theory.
		Abundant numbers, along with perfect, deficient, and amicable numbers, were thought to have mystical connections and played an important role in magic, sorcery, astrology, and horoscopes.
	www.hipfamily.com /archives/2006/04/wiki_wednesday_3.html (365 words)

	Almost perfect number (Site not responding. Last check: 2007-10-13)
		In mathematics, an almost perfect number (sometimes also called slightly defective number) is a natural number n such that the sum of all divisors of n (the divisor function σ(n)) is equal to 2n - 1.
		The only almost perfect numbers known are those of the form 2
		for some natural number k; however, it has not been shown yet that all almost perfect numbers are of this form.
	bopedia.com /en/wikipedia/a/al/almost_perfect_number.html (82 words)

	abundant number
		It is the square of 30 and a Harshad number.
		Twenty is a composite number, its proper divisors being 1, 2, 4, 5 and 10.
		210 is the sum of eight consecutive prime numbers (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210).
	www.experiencefestival.com /abundant_number (916 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: 2007-10-13)
		The first perfect number is 6, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6.
		Even perfect numbers have a very precise form; odd perfect numbers are rare, if indeed they do exist.
		There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's Strong Law of Small Numbers:
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=perfect_number (1268 words)

	Perfect, amicable and sociable numbers
		For a number n, we define s(n) to be the sum of the aliquot parts of n, i.e., the sum of the positive divisors of n, excluding n itself: so, for example, s(8)=1+2+4=7, and s(12)=1+2+3+4+6=16.
		A perfect number is a cycle of length 1 of s, i.e., a number whose positive divisors (except for itself) sum to itself.
		However the sum of unitary divisors, bi-unitary divisors, or infinitary divisors of a number is always even, unless the number is a power of 2.
	djm.cc /amicable.html (2519 words)

	Semiperfect number - Wikipedia, the free encyclopedia
		In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors.
		A semiperfect number that is equal to the sum of all its proper divisors is called a perfect number; an abundant number which is not semiperfect is called a weird number.
		With one exception, all primary pseudoperfect numbers are semiperfect.
	en.wikipedia.org /wiki/Semiperfect_number (159 words)

	[No title]
		Here are some recent papers on odd perfect numbers, from Math Reviews: [1] 1 651 762 (Review) Iannucci, Douglas E. The third largest prime divisor of an odd perfect number exceeds one hundred.
		Proceedings of the 5th Conference of the Canadian Number Theory Association held at Carleton University, Ottawa, ON, August 17--22, 1996.
		Proceedings of the Fourth Conference of the Canadian Number Theory Association held at Dalhousie University, Halifax, Nova Scotia, July 2--8, 1994.
	www.math.niu.edu /~rusin/known-math/00_incoming/odd_perf (878 words)

	Abundant Health -- Recommendations and Resources (Site not responding. Last check: 2007-10-13)
		Percent natural abundances refer to the relative proportions, expressed as percentages by number, in which the isotopes of an element are found in natural sources.
		Since most articles are very small (under 10k), and size, therefore, is not an issue, there is no valid reason to ''exclude'' material on the basis of its redundancy of external material alone.
		Unfortunately, I am unsure whether it is supposed to actually be a "closed interval" (which would match the notation), an "open interval" (in which case, I believe the bounds should be marked with parenthesis and not square brackets), or I am simply wrong.
	www.becomingapediatrician.com /health/1/abundant-health.html (1011 words)

	Abundant number: Encyclopedia - Abundant number
		An equivalent definition is that the proper divisors of the number (the divisors except the number itself) sum to more than the number.
		The first few abundant numbers (sequence A005101 in OEIS) are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66...
		Abundant number, Highly composite number, Quasiperfect number, Superabundant number
	www.experiencefestival.com /a/Abundant_number/id/416055 (541 words)

	Home > , {New York}, {NY}, __ZIP_CODE__, Real Estate, Yellow Pages, Classifieds, News, Events, Business, Shopping, ... (Site not responding. Last check: 2007-10-13)
		The first perfect number is 6, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6.
		Euclid discovered that the first four perfect numbers are generated by the form
		Curtiss (1922) uses a greedy algorithm for Egyptian fractions to prove that a perfect number N must have a number of divisors at least proportional to lnlnN-.
	www.centralparknyusa.com /section/Perfect_numbers (1367 words)

	Perfect Number (Site not responding. Last check: 2007-10-13)
		Odd perfect number must have at least six different prime aliquot factors (or eight if it is not divisible by 3; Ball and Coxeter 1987).
		Odd perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime aliquot factors.
		Odd perfect number must be a sum of squares.
	www.math.sdu.edu.cn /mathency/math/p/p240.htm (421 words)

	Quasiperfect Number -- from Wolfram MathWorld
		A quasiperfect number is a "least" abundant number, i.e., one such that
		No quasiperfect numbers are known, although if any exist, they must be greater than
		Singh (1997) called quasiperfect numbers slightly excessive numbers.
	mathworld.wolfram.com /QuasiperfectNumber.html (144 words)

	Imaginary number (Site not responding. Last check: 2007-10-13)
		666 is the 36th triangular number of complex numbers received a notable expansion.
		complex numbers are an algebraically closed field, fields the specified number as a long.
		© numbers is (11) where is the floor function amount of information is a dimensionless quantity.
	number.greatphonebook.net /their.html (727 words)

	Almost Perfect Number -- from Wolfram MathWorld
		An almost perfect number, also known as a least deficient or slightly defective (Singh 1997) number, is a positive integer
		The only known almost perfect numbers are the powers of 2, namely 1, 2, 4, 8, 16, 32,...
		It seems to be an open problem to show that a number is almost perfect iff it is of the form
	mathworld.wolfram.com /AlmostPerfectNumber.html (154 words)

	PPI no. 4, 1989
		+ 2, where r is the number of check symbols, then the check matrix is symmetric in the following sense: the matrix columns may be partitioned into N/2 pairs so that the sum of the columns in each pair is constant.
		If the conflict multiplicity is fixed and the number of transmitting stations n tends to infinity, then an algorithm with minimal (up to a constant multiplier) worst-case time is constructed in almost linear time O(nlog
		Each served customer is instantaneously routed with equal probability to one of M servers in the system (or is enqueued if the server is busy).
	www.ee.umd.edu /~abarg/ppi/contents/4-89-abstracts.html (710 words)

	IEEE Transactions on Computers
		While a certain resource in the hypercube may be shared by cube nodes to lower the cost, multiple copies of a shared resource often exist in the hypercube to reduce contention, and thus the potential delay, in fetching any shared copy.
		The number of increasing cycles for this new algorithm is only one, and the rounding result using this algorithm satisfies IEEE Standard 754 rounding perfectly.
		A number of errors have been discovered in the paper "Synthetic traces for trace-driven simulation of cache memories" by D. Thjebaut, J.L. Wolf and H.S. Stone.
	csdl2.computer.org /comp/trans/tc/1994/01/t1toc.xml (3025 words)

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