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 . 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.

## Contents

## 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 has solutions only if is a refactorable number, where is the greatest common divisor function.

Let be the number of refactorable numbers which are at most . The problem of determining an asymptotic for is open. Spiro has proven that ^{[2]}

There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large such that both and are refactorable. Zelinsky wondered if there exists a refactorable number , does there necessarily exist such that is refactorable and .

## History

First defined by Curtis Cooper and Robert E. Kennedy^{[3]} 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.^{[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