{{Short description|Integer divisible by the number of its divisors}} {{Redirect|Tau number|the ratio of a circle's circumference to its radius|Tau (mathematics)}} [[File:Refactorable number Cuisenaire rods 12.png|thumb|Demonstration, with Cuisenaire rods, that 1, 2, 8, 9, and 12 are refactorable]]
A '''refactorable number''' or '''tau number''' is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that <math>\tau(n)\mid n</math> with <math>\tau(n)=\sigma_0(n)=\prod_{i=1}^{n}(e_i+1)</math> for <math>n=\prod_{i=1}^np_i^{e_i}</math>. The first few refactorable numbers are listed in {{OEIS|id=A033950}} as :1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.
==Properties== 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. The equation <math>\gcd(n,x) = \tau(n)</math> has solutions only if <math>n</math> is a refactorable number, where <math>\gcd</math> is the greatest common divisor function.
Let <math>T(x)</math> be the number of refactorable numbers which are at most <math>x</math>. The problem of determining an asymptotic for <math>T(x)</math> is open. Spiro has proven that <math>T(x) = \frac{x}{\sqrt{\log x} (\log \log x)^{1-o(1)}}</math><ref>{{cite journal|last1=Spiro|first1=Claudia|title=How often is the number of divisors of n a divisor of n?|journal=Journal of Number Theory|date=1985|volume=21|issue=1|pages=81–100|doi=10.1016/0022-314X(85)90012-5|doi-access=free}}</ref>
There are still unsolved problems regarding refactorable numbers. Colton asked if there are arbitrarily large <math>n</math> such that both <math>n</math> and <math>n + 1</math> 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 <math>n</math> is refactorable and <math>n \equiv a \mod m</math>.
==History== First defined by Curtis Cooper and Robert E. Kennedy<ref> Cooper, C.N. and Kennedy, R. E. [https://dx.doi.org/10.1155/S0161171290000576 "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 have natural density zero, they were later rediscovered by Simon Colton using a computer program he wrote ("HR") 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.
==See also== {{Wikifunctions|Z15186|refactorable number checking}} * Divisor function
==References==
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{{Classes of natural numbers}} Category:Integer sequences