Refactorable number
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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 . The first few refactorable numbers are listed in (sequence A033950 in the OEIS) 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
[edit]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 has solutions only if is a refactorable number, where is the greatest common divisor function.
Let be the number of refactorable numbers which are at most . The problem of determining an asymptotic for is open. Spiro has proven that [2]
There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large such that both and are refactorable. Zelinsky wondered if there exists a refactorable number , does there necessarily exist such that is refactorable and .
History
[edit]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.
See also
[edit]References
[edit]- ^ J. Zelinsky, "Tau Numbers: A Partial Proof of a Conjecture and Other Results," Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8
- ^ Spiro, Claudia (1985). "How often is the number of divisors of n a divisor of n?". Journal of Number Theory. 21 (1): 81–100. doi:10.1016/0022-314X(85)90012-5.
- ^ Cooper, C.N. and Kennedy, R. E. "Tau Numbers, Natural Density, and Hardy and Wright's Theorem 437." Internat. J. Math. Math. Sci. 13, 383-386, 1990
- ^ S. Colton, "Refactorable Numbers - A Machine Invention," Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.2