Refactorable number

Properties

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) = \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)}}

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 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. 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

References

1. 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
2. (1985). "How often is the number of divisors of n a divisor of n?". Journal of Number Theory.
3. 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
4. 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