refactorable number

{{Short description|Integer divisible by the number of its divisors}}

{{Redirect|Tau number|the ratio of a circle's circumference to its radius|Turn (angle)#Proposals for a single letter to represent 2π}}

File:Refactorable number Cuisenaire rods 12.png, that 1, 2, 8, 9, and 12 are refactorable]]

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 $\tau(n)\mid n$ with $\tau(n)=\sigma_0(n)=\prod_{i=1}^{n}(e_i+1)$ for $n=\prod_{i=1}^np_i^{e_i}$ . The first few refactorable numbers are listed in {{OEIS|id=A033950}} 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.J. Zelinsky, "[http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.pdf Tau Numbers: A Partial Proof of a Conjecture and Other Results]," Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8 Colton proved that no refactorable number is perfect. The equation $\gcd(n,x) = \tau(n)$ has solutions only if $n$ is a refactorable number, where $\gcd$ is the greatest common divisor function.

Let $T(x)$ be the number of refactorable numbers which are at most $x$ . The problem of determining an asymptotic for $T(x)$ is open. Spiro has proven that $T(x) = \frac{x}{\sqrt{\log x} (\log \log x)^{1-o(1)}}$ {{cite journal|last1=Spiro|first1=Claudia|title=How often is the number of divisors of n a divisor of n?|journal=Journal of Number Theory|date=1985|volume=21|issue=1|pages=81–100|doi=10.1016/0022-314X(85)90012-5|doi-access=free}}

There are still unsolved problems regarding refactorable numbers. Colton asked if there are arbitrarily large $n$ such that both $n$ and $n + 1$ are refactorable. Zelinsky wondered if there exists a refactorable number $n_0 \equiv a \mod m$ , does there necessarily exist $n > n_0$ such that $n$ is refactorable and $n \equiv a \mod m$ .

History

First defined by Curtis Cooper and Robert E. Kennedy Cooper, C.N. and Kennedy, R. E. [https://dx.doi.org/10.1155/S0161171290000576 "Tau Numbers, Natural Density, and Hardy and Wright's Theorem 437."] Internat. J. Math. Math. Sci. 13, 383-386, 1990 where they showed that the tau numbers have natural density zero, they were later rediscovered by Simon Colton using a computer program he wrote ("HR") which invents and judges definitions from a variety of areas of mathematics such as number theory and graph theory.S. Colton, "[http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html Refactorable Numbers - A Machine Invention]," Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.2 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.

References

Category:Integer sequences

refactorable number

Properties

History

See also

References