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Topic: Niven numbers

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Harshad number

48 (number)

Bernoulli numbers

Book of Numbers

Larry Niven

Elkhart, Texas

Daniel Parker

The Forge of God

United States

	The Sound of Mathematics - Numbers
		A Niven number (or Harshad number, but I give priority to alliterations) is a positive integer which is divisible by the sum of its digits.
		This value is the number of tied sixteenth notes and the number of scale steps relative to the previous tone.
		When it's done, it hooks up at the prime number where the 1st part is at the monent (I forgot the exact number) and uses the same algorithm as the 2nd part had at the beginning of the piece.
	www.geocities.com /Vienna/9349/numbers.html (677 words)

	PlanetMath: Harshad number
		All 1-digit numbers and the base number itself are Harshad numbers.
		It is possible for an integer to be divisible by its digital root and yet not be a Harshad number because it doesn't divide its first digit sum evenly (for example, 38 in base 10 has digital root 2 but is not divisible by 3 + 8 = 11).
		This is version 2 of Harshad number, born on 2006-03-18, modified 2006-06-13.
	planetmath.org /encyclopedia/NivenNumber.html (177 words)

	Niven number
		For example, 126 is a Niven number because, the sum of its digits 1 + 2 + 6, is 9, and 9 goes into 126 exactly 14 times.
		Niven numbers are name after the number theorist Ivan Niven who, in 1977, gave a talk at a conference in which he mentioned integers which are twice the sum of their digits.
		Then in a 1982 article, the mathematician Robert Kennedy christened numbers which are divisible by their digital sum in honor of Niven.
	www.daviddarling.info /encyclopedia/N/Niven_number.html (165 words)

	Larry Niven
		Many of Niven's stories take place in his Known Space universe, in which humanity shares the several solar systems nearest to Sol with over a dozen alien species, including species known as the Kzinti, and Pierson's Puppeteers, which are frequently central characters.
		Niven's idea of a beanstalk sucking dry a planet (see Rainbow Mars) seems to be copied in the animated movie Kaena: The Prophecy and also in the game Halo 2 by game company Bungie.
		Larry Niven introduced the idea of a flash crowd in his story Flash Crowd, 1973, which evolved in 2003 to the flash mob in which people meet together to protest in a creative way at a specific time and place to disappear as quickly as they appeared some minutes later.
	www.sfcrowsnest.com /scifinder/a/Larry_Niven.php (867 words)

	Harshad number
		For example, 1729 is a Harshad number because 1 + 7 + 2 + 9 = 19 and 1729 = 19 x 91.
		A Harshad amicable pair is an amicable pair (m, n) such that both m and n are Harshad numbers.
		For example, 2620 and 2924 are a Harshad amicable pair because 2620 is divisible by 2 + 6 + 2 + 0 = 10 and 2924 is divisible by 2 + 9 + 2 + 4 = 17 (2924/17 = 172).
	www.daviddarling.info /encyclopedia/H/Harshad_number.html (184 words)

	MATHEWS: Niven Numbers (Site not responding. Last check: 2007-10-11)
		Niven numbers (also called Harshad numbers) are positive integers which are divisible by the sum of the digits.
		Niven numbers are named by Robert E. Kennedy in honor of I. Niven who mentioned them 1977 in a conference on Number Theory.
		Niven numbers are connected to Smith numbers in the way that they overlap infinitely often.
	www.wschnei.de /digit-related-numbers/niven-numbers.html (318 words)

	Math Awareness Day 2001
		A Niven number is a positive integer that is divisible by the sum of its digits.
		An analog to the concept of a Niven number is that of an n-Niven number.
		An n-Niven number is a number divisible by the sum of the digits in its base n expansion.
	www.sju.edu /~rhall/mathday.html (175 words)

	Niven numbers
		Niven numbers (also called Harshad numbers or multidigital numbers) are positive integers divisible by the sum of their digits in base 10.
		For any natural number n>0, one may also define n-Niven numbers to be those positive integers which are divisible by the sum of their digits in base n.
		Trivially, the numbers up to n-1 are n-Niven numbers (since they are single-digit, the sum of digits is themselves), and n is an n-Niven number (since n in base n is always 10).
	www.mik.fastload.org /ni/Niven_numbers.html (180 words)

	Ivan Niven
		Ivan Niven was born in Vancouver, B. on October 25, 1915.
		Ivan Niven has been the complete mathematician who was noted for outstanding teaching, popular books, a life-long active research program, and generous service to general mathematics community.
		His areas of expertise were number theory, especially the areas of diophantine approximation and questions of irrationality and transcendance of numbers, and combinatorics.
	www.numbertheory.org /obituaries/OTHERS/niven/nivenobit.html (1163 words)

	Harshad number - Wikipedia, the free encyclopedia
		A Harshad number, or Niven number, is an integer that is divisible by the sum of its digits in a given number base.
		For example, 99, although divisible by 9 as shown by 9 + 9 = 18 and 1 + 8 = 9, is not a Harshad number, since 9 + 9 = 18, and 99 is not divisible by 18.
		For a prime number to also be a Harshad number, it must be less than the base number, (that is, a 1-digit number) or the base number itself.
	en.wikipedia.org /wiki/Harshad_number (698 words)

	Normal Numbers (Site not responding. Last check: 2007-10-11)
		A normal number is a real number for which the representation in any base has the property that any block of consecutive digits is equally likely to occur as we move along the digit expansion of the number.
		It is a little surprising at first that almost all real numbers have this property; however, it is quite a chore to come up with even one specific example.
		The project would be to summarize why almost all numbers are normal and to work through a proof that a specific number is normal.
	www.math.okstate.edu /~wrightd/4713/papers/node6.html (147 words)

	Read This: Maxima and Minima Without Calculus
		Niven explores isoperimetric problems for triangles, quadrilaterals, and inscribed and circumscribed polygons, as well as varrious other topics, such as trigonometry, ellipses, and Euclidean 3-space.
		What is most striking about this book is that Niven shows how a proof that assumes a solution exists is incomplete because it doesn´t prove that a maximal or minimal solution must exist.
		Niven intends Maxima and Minima to serve as a "resource book, not a textbook" because there are some problems left for the reader to solve, but not that many.
	www.maa.org /reviews/MaxMinNiven.html (496 words)

	Fascinating Smith Numbers by Shyam Sunder Gupta
		Numbers such that s(n), the sum of aliquot divisors of n, is greater than n are called Abundant numbers.
		Numbers such that s(n), the sum of aliquot divisors of n, is less than n are called Deficient numbers.
		as the numbers for which the sum of the digits of the prime factors is equal to k multiplied by sum of the digits i.e.
	www.shyamsundergupta.com /smith.htm (1983 words)

	MathTrek: Niven Numbers (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-11)
		A Niven number is an integer divisible by the sum of its digits.
		For example, 476 is a Niven number because the sum of its digits is 17 and 17 divides evenly into 476.
		The first few Niven numbers with more than one digit in base 10 are: 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100.
	blog.sciencenews.org.cob-web.org:8888 /2006/03/niven_numbers.html (527 words)

	Neutron Star Still Shining Light on Hard Science Fiction
		Though Larry Niven is best known for the Hugo- and Nebula-winning novel Ringworld, Neutron Star is in many ways an even more seminal book, both to Niven's Known Space future history and to SF in general.
		Niven ensured that alien technologies were unique to their native species' psychology and physiology, crafting yet another facet of a thoroughly believable whole.
		If the author is lucky, his or her short work might be picked up for publication by a small press or anthologized in Gardner Dozois' or David Hartwell's annual "Best of" anthologies -- otherwise it will languish in the pages of the magazine in which it was originally published, forever out of print.
	www.space.com /sciencefiction/larryniven/neutron_star_000217.html (1238 words)

	MathTrek: Number Gossip
		So we have numbers that are perfect, amicable, lucky, narcissistic, weird, and so on.
		Moreover, 720 is the largest factorial to contain all different digits, and 720 is the smallest number with 30 divisors.
		By the way, a number is evil if it has an even number of 1s in its binary representation.
	blog.sciencenews.org /2006/07/number_gossip_2.html (283 words)

	Amicable Numbers by Shyam Sunder Gupta
		The numbers 220 and 284 form the smallest pair of amicable numbers (also known as friendly numbers) known to Pythagoras.
		Two numbers are called Amicable (or friendly) if each equals to the sum of the aliquot divisors of the other (aliquot divisors means all the divisors excluding the number itself).
		Harshad (or Niven) numbers are those numbers which are divisible by their sum of the digits.
	www.shyamsundergupta.com /amicable.htm (1129 words)

	Ivars Peterson's MathTrek
		Whole numbers or integers are often the subject of such pursuits.
		Interestingly, when the digits of the original number are added together, the result (42) equals the sum of the digits of the prime factors (3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42).
		A commentary on Smith (and Rhonda) numbers, along with a discussion of their mathematical usefulness, is available at http://www.seanet.com/~ksbrown/kmath007.htm.
	www.maa.org /mathland/mathtrek_10_27.html (783 words)

	Math Trek: Names for Numbers, Science News Online, July 29, 2006
		Then, of course, there are Rhonda numbers, named by Kevin Brown (see http://www.mathpages.com/home/kmath007.htm) in honor of an acquaintance whose address included the number 25662.
		So, a Rhonda number in base b is a number such that the product of the digits in base b equals b times the sum of its prime factors (b itself must be composite).
		There's also a chance that the study of numbers expressed in a certain base highlights something unusual in the realm of mathematics—something that distinguishes numbers expressed in one base from those expressed in another.
	www.sciencenews.org /articles/20060729/mathtrek.asp (1113 words)

	G4 - Feature - Larry Niven Interview Prep
		Niven was gracious enough to write us a quick essay on the scientific theories -- proven and disproven -- that have influenced his life and his work:
		Niven numbers, also known as Harshad or multidigital numbers, are widely thought to have been named in the author's honor.
		The numbers are defined as those positive integers that are divisible by the sum of their digits.
	www.g4tv.com /screensavers/features/41876/Larry_Niven_Interview_Prep.html (622 words)

	diophanfin.html
		Diophantus stated the traditional definition of a number to be a collection of units, but in his problems, he referred to each of his positive rational solutions as a number (Bashmakova 5).
		Also, since we can divide a rational number by another rational and always have a rational as an answer (since, that's right, the rationals are a field) this is equivalent to finding the rational points on the unit circle.
		Fermat's work with prime numbers is the basis of modern encryption techniques, which is actually based on Euler's generalization of Fermat's prime number theories.
	www.ms.uky.edu /~carl/ma330/projects/diophanfin1.html (2434 words)

	[No title]
		One can go one step farther and say it is the one that brought irratoinal numbers into the realm of mathematics.The story is very short.Using our modern terminology, the Phythagoreans thought that all numbers are rational.They were shocked when they found that the length of the diagonal of the unit square is not rational.
		For example, the numbers $\alpha =.a_{1}a_{2}a_{3}...$, where $a_{i}=1$ if i is prime and 0 otherwise, and $\beta =.p_{1}p_{2}p_{3}...$, where $p_{i}$ is the sequence of primes given in increasing order, are irrational.
		There are some numbers whose irrationality or transcendency are not yet known.
	www.math.rutgers.edu /~zeilberg/essays683/rashidi (732 words)

	Francis L. Niven Papers, 1901-1982 (Collection 397)
		Francis Niven was born and raised in the Gallatin Valley.
		Niven married Ethel Scheytt, a former resident of Maudlow, Montana and the couple continued operating the North Side Market while Francis attended college during the Depression.
		Photographs of Francis Niven and his prize winning sheep shown at various gatherings including the Montana Winter Fair; photographs of the 1958 shipment of livestock to Turkey; snapshots of unidentified people in the Maudlow, Montana area probably taken by Ethel Scheytt Niven, circa 1920s.
	www.lib.montana.edu /collect/spcoll/findaid/0397.html (813 words)

	18.100B Fall 2002 (Site not responding. Last check: 2007-10-11)
		The fact that this class tries to be non-algebraic makes it seem a little silly at times; they are clearly struggling not to explain things in the more concise language of abstract algebra, but this class can be seen as a useful introduction to algebra and some of its applications.
		In a class meant to survey and provoke interest in number theory, he seemed to get excited and digress on things that a person with a working knowledge of number theory might find interesting, but that the novice might find stupid.
		The time when he used modular arithmetic to find the ninth root of a 30 digit number was impressive and really showed the students why we were learning this archaic stuff.
	web.mit.edu /uma/www/ug/18.781s3.html (515 words)

	Niven Numbers
		A Niven Number is any whole number that is divisible by the sum of its digits.
		They seem to be overly concerned about being unable to state the number of Niven Numbers in that range in a short amount of time, rather than realizing that it will take some time and patience to arrive at it.
		In 1977, Ivan Niven, a famous number theorist presented a talk at a conference in which he mentioned integers which are twice the sum of their digits.
	www.trottermath.net /numthry/nivennos.html (1645 words)

	Special Numbers I
		Believe it or not, but it is the smallest number that is expressable as the sum of two 4th powers in two different ways.
		If you select any prime number, greater than 3, square it, then diminish that by 1, then 24 is always a divisor (factor) of the result.
		Probably you recognize this number; it's the 6-digit period of the rational number 1/7.
	www.trottermath.net /numtrivia/specnum.html (1287 words)

	MathTrek: Uninteresting Numbers, Not
		One of the many delights of the book From Zero to Infinity: What Makes Numbers Interesting by Constance Reid is a little "proof" that there are no uninteresting numbers.
		The natural numbers, which are the primary subject of this book, do not end with the digits with which we represent them.
		And they are all interesting: for if there were any uninteresting numbers, there would of necessity be a smallest uninteresting number and it, for that reason alone, would be very interesting.
	blog.sciencenews.org /2006/03/uninteresting_numbers_not.html (197 words)

	Amazon.com: Irrational Numbers (Carus Monograph): Books: Ivan Niven (Site not responding. Last check: 2007-10-11)
		The approximation of irrational numbers by rationals, up to such results as the best possible approximation of Hurwitz, is also given with elementary technique.
		Ivan Niven earned his Bachelor's and Master's degrees (1934 and 1936) at the University of British Columbia, and he was awarded his Ph.D. in 1938 at the University of Chicago.
		Ivan Niven died in May 1999 in Eugene, Oregon --This text refers to the Paperback edition.
	www.amazon.com /Irrational-Numbers-Carus-Monograph-Niven/dp/0883850117 (969 words)

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