# Refactorable number

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]\displaystyle{ \tau(n)\mid n }[/math]. The first few refactorable numbers are listed in (sequence A033950 in the OEIS) 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.^{[1]} Colton proved that no refactorable number is perfect. The equation [math]\displaystyle{ \gcd(n,x) = \tau(n) }[/math] has solutions only if [math]\displaystyle{ n }[/math] is a refactorable number, where [math]\displaystyle{ \gcd }[/math] is the greatest common divisor function.

Let [math]\displaystyle{ T(x) }[/math] be the number of refactorable numbers which are at most [math]\displaystyle{ x }[/math]. The problem of determining an asymptotic for [math]\displaystyle{ T(x) }[/math] is open. Spiro has proven that [math]\displaystyle{ T(x) = \frac{x}{\sqrt{\log x} (\log \log x)^{1+o(1)}} }[/math]^{[2]}

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

## History

First defined by Curtis Cooper and Robert E. Kennedy^{[3]} where they showed that the tau numbers have 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.^{[4]} 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

## References

- ↑ J. Zelinsky, "Tau Numbers: A Partial Proof of a Conjecture and Other Results,"
*Journal of Integer Sequences*, Vol. 5 (2002), Article 02.2.8 - ↑ Spiro, Claudia (1985). "How often is the number of divisors of n a divisor of n?".
*Journal of Number Theory***21**(1): 81–100. doi:10.1016/0022-314X(85)90012-5. - ↑ 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
- ↑ S. Colton, "Refactorable Numbers - A Machine Invention,"
*Journal of Integer Sequences*, Vol. 2 (1999), Article 99.1.2

Original source: https://en.wikipedia.org/wiki/Refactorable number.
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