Refactorable number

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 ${\displaystyle \gcd(n,x)=\tau (n)}$ has solutions only if ${\displaystyle n}$ is a refactorable number, where ${\displaystyle \gcd }$ is the greatest common divisor function.

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

There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large ${\displaystyle n}$ such that both ${\displaystyle n}$ and ${\displaystyle n+1}$ are refactorable. Zelinsky wondered if there exists a refactorable number ${\displaystyle n_{0}\equiv a\mod m}$, does there necessarily exist ${\displaystyle n>n_{0}}$ such that ${\displaystyle n}$ is refactorable and ${\displaystyle n\equiv a\mod m}$.

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.

• Divisor function