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 $\tau (n)|n$. The first few refactorable numbers are listed in (sequence A033950 in the OEIS) 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96

History

First defined by Curtis Cooper and Robert E. Kennedy 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[1]. Colton called such numbers "refactorable" While computer programs had discovered proofs before, this disovery 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.

Known properties

• Tau numbers have natural density zero (Cooper and Kennedy).
• No three consecutive integers are all tau (Zelinsky) [2]
• No tau number is perfect (Colton)
• The equation GCD(n, x) = τ(n) has solutions only if n is a refactorable number.

Unsolved problems

• Are there arbitrarily large $n$ such that both $n$ and $n+1$ are tau? (Conjectured by Colton)
• If there exists a tau number $n_{0}$ &equiv& $a$ (mod m), does there necessarily exist $n>n_{0}$ such that $n$ is tau and $n\equiv a$ (mod m) (Conjectured by Zelinsky)