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 ${\displaystyle \tau (n)|n}$. The first few refactorable numbers are listed in (sequence in the OEIS) as

For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6.

Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable. Colton proved that no refactorable number is perfect. The equation GCD(n, x) = τ(n) has solutions only if n is a refactorable number.

There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large n such that both n and 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 n is refactorable and ${\displaystyle n\equiv a\mod m}$.

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