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Topic: Abundant number

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	tingilinde: take a number ...
		22 is the number of partitions of 8.
		101 is the number of partitions of 13.
		231 is the number of partitions of 16.
	tingilinde.typepad.com /starstuff/2005/11/significant_int.html (12385 words)

	What's Special About This Number?
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.stetson.edu /~efriedma/numbers.html (7326 words)

	abundant number
		Twelve is the smallest abundant number – the sum of its aliquot parts is 1 + 2 + 3 + 4 + 6 = 16 – followed by 18, 20, 24, and 30.
		A weird number is an abundant number that is not semiperfect; in other words, n is weird if the sum of its divisors is greater than n, but n is not equal to the sum of any subset of its divisors.
		A number that is not abundant or deficient is known as a perfect number.
	www.daviddarling.info /encyclopedia/A/abundant_number.html (235 words)

	Abundant number - Wikipedia, the free encyclopedia
		In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n.
		Marc Deléglise showed in 1998 that the natural density of abundant numbers is between 0.2474 and 0.2480.
		An abundant number which is not a semiperfect number is called a weird number; an abundant number with abundance 1 is called a quasiperfect number.
	en.wikipedia.org /wiki/Abundant_number (288 words)

	Brewer, E. Cobham. Dictionary of Phrase & Fable. Abundant Number (An).
		A number such that the sum of all its divisors (except itself) is greater than the number itself.
		Thus 12 is an abundant number, because its divisors, 1, 2, 3, 4, 6 = 16, which is greater than 12.
		A Perfect number is one of which the sum of all its divisors exactly measures itself, as 6, the divisors of which are 1, 2, 3 = 6.
	www.bartleby.com /81/95.html (145 words)

	PlanetMath: abundant number
		Given a pair of amicable numbers, the lesser of the two is abundant.
		Empirical proof that every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers
		This is version 2 of abundant number, born on 2006-04-25, modified 2006-04-26.
	planetmath.org /encyclopedia/AbundantNumber.html (108 words)

	I (Site not responding. Last check: 2007-10-20)
		An abundant number is a number that the factors add above the given number (except 1 and the given number).
		A deficient number is a number that when you add up all the factors of the number, the factors add up to be less than the given number.
		A Perfect number is defined as a number whose proper factors add up to be that number.6 is the first Perfect number.
	www.promotega.org /vsu05003/essay1.htm (584 words)

	Multiperfect numbers: Glossary
		An abundant number which is neither a multiple of another abundant number nor a multiple of a perfect number is called "primitive abundant".
		A number whose index is an integer; equivalently, a number such that the sum of its divisors is a multiple of the number.
		A number is proper multiperfect if its index is an integer greater than two; i.e., if it is a multiperfect number greater than one which is not perfect.
	pw1.netcom.com /~fredh/mpfn/glossary.html (941 words)

	The Prime Glossary: abundant number
		There are infinitely many abundant numbers, both even (e.g., every multiple of 12) and odd (e.g., every odd multiple of 945).
		Every proper multiple of a perfect number, and every multiple of an abundant number, is abundant (because when n > 1, sigma(n)/n > 1+1/n; and sigma is a multiplicative function).
		Deleglise has shown that on the average 24.7% of the positive integers are abundant (more specifically, that the natural density of the abundant integers is in the open interval (0.2474, 0.2480)).
	primes.utm.edu /glossary/page.php?sort=AbundantNumber (178 words)

	Perfect Numbers?--Mathematics: Grades 5-8 (Site not responding. Last check: 2007-10-20)
		The proper factors of a number are all of the factors of a number except the number itself.
		Numbers may be designated perfect, defective, or abundant based on the sum of their proper factors.
		Abundant numbers are those numbers where the sum of the proper factors is greater than the number itself.
	www.teachercreated.com /lessons/001201cm.shtml (617 words)

	Puzzle 329. Odd abundant numbers not divided by 2 or 3.
		There are 26 abundant numbers not divisible by 2 or 3 and less than 10^11, from 5391411025 to 97974952075.
		Regarding your puzzle 329 "Odd abundant numbers not divided by 2 or 3", It is to inform that there are many many numbers known.
		On my site (www.shyamsundergupta.com/canyoufind.htm)in reply to CYF No. 7, Brian Trial reported three consecutive abundant numbers two of which are odds the number 27523728059933744479478128774219545978059153664131777 is abundant number not divisible by 6.
	www.primepuzzles.net /puzzles/puzz_329.htm (926 words)

	Puzzle 233. A little twist
		Find the earliest two solutions for numbers n such that the sum of the proper divisors of n are greater than n and no subset of these divisors sums up to n.
		6 is a perfect number and we know that: " All number which is a multiple of a perfect number or of an abundant number is an abundant number " (It is easy to show it) Thus all even pandigital number is an abundant number.
		1°) Among the odd numbers of the form a^nb^mp with 3<=anumbers), the only ones that are abundant numbers are of the form : 3^n5^m7 or 3^n5^m11 or 3^n5^m13.
	www.primepuzzles.net /puzzles/puzz_233.htm (1550 words)

	Colossally abundant number - Wikipedia, the free encyclopedia
		In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number.
		The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040,...
		Keith Briggs on colossally abundant numbers and the Riemann hypothesis
	en.wikipedia.org /wiki/Colossally_abundant_number (235 words)

	High School Problems Archive (Site not responding. Last check: 2007-10-20)
		An abundant number is a positive integer, the sum of whose distinct proper factors is greater than the number.
		The ones with a sum that is larger than the number are abundant.
		No prime numbers appear in the table, since the only proper factor of a prime number is 1.
	www.nctm.org /high/asolutions.asp?ID=414 (117 words)

	Perfect Number
		Euclid was one of the first to study perfect numbers, although, due to the difficulty of finding such numbers, he only knew of the first four perfect numbers: 6, 28, 496, and 8128.
		A 'deficient' number is, as one might guess, a number whose proper divisors sum to less than twice itself.
		In his book Perfect Numbers, Richard Shoemaker defines social numbers "as being a chain of numbers in which each number is equal to the sum of all the proper divisors of the preceding number, the last being considered as preceding the first number of the chain" (page 27).
	math.arizona.edu /~ura/001/gaberdiel.jw (6796 words)

	Exam 1 Review:
		The number of partitions found is p(12, 3) + p(12,2) + p(12,1).
		Explain why 7 is the largest prime number needed to find the primes less than 50.
		n is assumed to be a composite number.
	home.snu.edu /~lturner/MC-MathStr/ReviewExam1.htm (328 words)

	Math 117 - Section 91
		An abundant number may be a prime number.
		Goldbach’s Conjecture states that every even number greater than 2 can be expressed as the sum of two prime numbers.
		The seventeenth Fibonacci number is 1597 and the eighteenth Fibonacci number is 2584.
	www2.smumn.edu /facpages/~wlarson/100ABCTEST1solns.htm (406 words)

	C++ Looping Issue - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-10-20)
		Once it takes that number, it has to determine whether it is a deficient, perfect, or abundant number based on the sum of its divisors(or factors).
		1, 2, 3, 4, 6 (the number itself, in this case 12, is exempt).
		Therefore, 16 > 12 and so 12 is considered an "abundant" number since the sum of its factors is greater than the number itself.
	www.gamedev.net /community/forums/viewreply.asp?ID=2501474 (1376 words)

	On the Number 85 (Part 1)
		Sum of the 14th prime & 8th abundant number = 43 + 42 = 85
		Sum of the 6th prime numbers & 15th abundant number = 13 + 72 = 85
		The characters live at number 85 in the kookiest block of flats in Britain.
	www.wisdomportal.com /Numbers/85-1.html (7046 words)

	Abundant - definition from Biology-Online.org
		(Science: mathematics) abundant number, a number, the sum of whose aliquot parts exceeds the number itself.
		thus, 1, 2, 3, 4, 6, the aliquot parts of 12, make the number 16.
		this is opposed to a deficient number, as 14, whose aliquot parts are 1, 2, 7, the sum of which is 10; and to a perfect number, which is equal to the sum of its aliquot parts, as 6, whose aliquot parts are 1, 2.
	www.biology-online.org /dictionary/Abundant (180 words)

	On the Number 54 (Part 1)
		Sum of the 1st and 8th abundant numbers = 12 + 42 = 54
		Sum of the 2nd and 6th abundant numbers = 18 + 36 = 54
		Sum of the 4th and 5th abundant numbers = 24 + 30 = 54
	www.wisdomportal.com /Numbers/54-1.html (6259 words)

	View from Number 80- Abundant Pendulum Mysteries - Skeptical Reviews
		Which is why 80 is reluctantly passing up Unlimited Financial Abundance - better to keep the money you have than give it to the likes of Glen Harrold.
		- Whilst trawling through a few other audio self-help sites (of which there are a depressing number) one that popped up, the Pendulum Warehouse, introduced the concept of using a pendulum in conjunction with your tapes for greater benefit.
		Here the subject of pendulums is covered in great detail - most of it unverifiable as no references are supplied.
	www.number80.co.uk /page56.htm (3307 words)

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