Zero-stage - Revision history

en>Yobot: Tagging, added underlinked tag using AWB (10458)

2014-09-16T07:03:58Z

Tagging, added underlinked tag using AWB (10458)

← Older revision		Revision as of 09:03, 16 September 2014
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	~~A '''refactorable number''' or '''tau number'''~~ is ~~an integer~~ '~~'n'' that is divisible by~~ the ~~count of its [[divisor]]s, or~~ to ~~put it algebraically, ''n'' is such that <math>\tau(n)\|n</math>~~. The first few refactorable numbers are listed in {{OEIS\|id=A033950}} [[1 (number)\|1]], [[2 (number)\|2]], [[8 (number)\|8]], [[9 (number)\|9]], [[12 (number)\|12]], [[18 (number)\|18]], [[24 (number)\|24]], [[36 (number)\|36]], [[40 (number)\|40]], [[56 (number)\|56]], [[60 (number)\|60]], [[72 (number)\|72]], [[80 (number)\|80]], [[84 (number)\|84]], [[88 (number)\|88]], [[96 (number)\|96]]. For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is ~~divisible by 6.~~		Myrtle Benny is how I'm called and I feel comfy when people use the full name. Years ago we moved to North Dakota. To collect coins is what his family members and him enjoy. For many years I've been working as a payroll clerk.<br><br>Also visit my page :: [http://www.lankaclipstv.com/blog/165665 http://www.lankaclipstv.com/]

	~~Cooper~~ and ~~Kennedy proved that refactorable numbers have [[natural density]] zero~~. Zelinsky proved that no three consecutive integers can all be refactorable.<ref>J. Zelinsky, "[http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.pdf Tau Numbers: A Partial Proof of a Conjecture and Other Results]," ''Journal of Integer Sequences'', Vol. 5 (2002), Article 02.2.8</ref> Colton proved that no refactorable number is [[perfect number\|perfect]]. The equation [[greatest common divisor\|GCD]](''n'', ''x'') = τ(''n'') has solutions only if ''n'' is a ~~refactorable number.~~

	~~There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large ''n'' such that both ''n'' and ''n'' + 1 are refactorable~~. ~~Zelinsky wondered if there exists a refactorable number~~ <~~math~~>~~n_0 \equiv a \mod m~~<~~/math~~>~~, does there necessarily exist <math>n > n_0</math> such that ''n'' is refactorable and <math>n \equiv a \mod m</math>.~~

	~~==History==~~

	First defined by [[Curtis Cooper (mathematician)\|Curtis Cooper]] and Robert E. Kennedy<ref> Cooper, C.N. and Kennedy, R. E. "Tau Numbers, Natural Density, and Hardy and Wright's Theorem 437." Internat. J. Math. Math. Sci. 13, 383-386, 1990 </ref> where they showed that the tau numbers has [[natural density]] zero, they were later rediscovered by [[Simon Colton]] using a computer program he had made which invents and judges definitions from a variety of areas of mathematics such as [[number theory]] and [[graph theory]].<ref>S. Colton, "[http://www.cs.~~uwaterloo.ca~~/~~journals~~/~~JIS~~/~~colton~~/~~joisol~~.~~html Refactorable Numbers - A Machine Invention]," ''Journal of Integer Sequences'', Vol~~. ~~2 (1999), Article 99.1.2<~~/ref> Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.

	~~==References==~~

	~~<references/>~~

	~~{{Classes of natural numbers}}~~
	~~[[Category:Integer sequences]~~]


	~~{{numtheory-stub}}~~

en>Dawynn: stub sorting using AWB

2013-12-28T11:31:30Z

stub sorting using AWB

← Older revision		Revision as of 13:31, 28 December 2013
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	~~The name~~ of ~~the writer~~ is ~~Jayson~~. ~~Alaska~~ is ~~exactly where he~~'~~s usually been residing~~. ~~My working day occupation~~ is ~~an information officer but I~~'~~ve already utilized for an additional one~~. ~~To climb~~ is ~~some thing she would~~ by ~~no means give up~~.<br><br>~~Feel free to visit my web page; online psychic reading (~~[~~https~~://www.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 machlitim~~.~~org~~.il])		A '''refactorable number''' or '''tau number''' is an integer ''n'' that is divisible by the count of its [[divisor]]s, or to put it algebraically, ''n'' is such that <math>\tau(n)\|n</math>. The first few refactorable numbers are listed in {{OEIS\|id=A033950}} [[1 (number)\|1]], [[2 (number)\|2]], [[8 (number)\|8]], [[9 (number)\|9]], [[12 (number)\|12]], [[18 (number)\|18]], [[24 (number)\|24]], [[36 (number)\|36]], [[40 (number)\|40]], [[56 (number)\|56]], [[60 (number)\|60]], [[72 (number)\|72]], [[80 (number)\|80]], [[84 (number)\|84]], [[88 (number)\|88]], [[96 (number)\|96]]. For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6.

			Cooper and Kennedy proved that refactorable numbers have [[natural density]] zero. Zelinsky proved that no three consecutive integers can all be refactorable.<ref>J. Zelinsky, "[http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.pdf Tau Numbers: A Partial Proof of a Conjecture and Other Results]," ''Journal of Integer Sequences'', Vol. 5 (2002), Article 02.2.8</ref> Colton proved that no refactorable number is [[perfect number\|perfect]]. The equation [[greatest common divisor\|GCD]](''n'', ''x'') = τ(''n'') has solutions only if ''n'' is a refactorable number.

			There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large ''n'' such that both ''n'' and ''n'' + 1 are refactorable. Zelinsky wondered if there exists a refactorable number <math>n_0 \equiv a \mod m</math>, does there necessarily exist <math>n > n_0</math> such that ''n'' is refactorable and <math>n \equiv a \mod m</math>.

			==History==

			First defined by [[Curtis Cooper (mathematician)\|Curtis Cooper]] and Robert E. Kennedy<ref> Cooper, C.N. and Kennedy, R. E. "Tau Numbers, Natural Density, and Hardy and Wright's Theorem 437." Internat. J. Math. Math. Sci. 13, 383-386, 1990 </ref> where they showed that the tau numbers has [[natural density]] zero, they were later rediscovered by [[Simon Colton]] using a computer program he had made which invents and judges definitions from a variety of areas of mathematics such as [[number theory]] and [[graph theory]].<ref>S. Colton, "[http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html Refactorable Numbers - A Machine Invention]," ''Journal of Integer Sequences'', Vol. 2 (1999), Article 99.1.2</ref> Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.

			==References==

			<references/>

			{{Classes of natural numbers}}
			[[Category:Integer sequences]]


			{{numtheory-stub}}

en>Jeepday: Wikipedia:CiterSquad; you can help!

2009-11-01T14:55:49Z

Wikipedia:CiterSquad; you can help!

New page

The name of the writer is Jayson. Alaska is exactly where he's usually been residing. My working day occupation is an information officer but I've already utilized for an additional one. To climb is some thing she would by no means give up.<br><br>Feel free to visit my web page; online psychic reading ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 machlitim.org.il])