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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.
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| 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.
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| 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>.
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| ==History==
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| 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.
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| ==References==
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| <references/>
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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| {{numtheory-stub}}
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