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 τ(n) | n. The first few refactorable numbers are listed in (sequence A033950 in OEIS) 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96 In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... Look up one in Wiktionary, the free dictionary. ... 2 (two) is a number, numeral, and glyph. ... 8 (eight) is the natural number following 7 and preceding 9. ... For other senses of this word, see 9 (disambiguation). ... 12 (twelve) is the natural number following 11 and preceding 13. ... 18 (eighteen) is the natural number following 17 and preceding 19. ... 24 (twenty-four) is the natural number following 23 and preceding 25. ... 36 is the natural number following 35 and preceding 37. ... 40 (forty) is the natural number following 39 and preceding 41. ... 56 (fifty-six) is the natural number following 55 and preceding 57. ... 60 (sixty) is the natural number following 59 and preceding 61. ... 72 is the natural number following 71 and preceding 73. ... 80 (eighty) is the natural number following 79 and preceding 81. ... 84 (eighty-four) is the natural number following 83 and preceding 85. ... 88 is the natural number following 87 and preceding 89. ... 96 is the natural number following 95 and preceding 97. ...

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 GCD(n, x) = τ(n) has solutions only if n is a refactorable number. In mathematics, a sequence a1, a2, ... , an, with the aj positive integers and aj < aj+1 for all j, has natural density α, where 0 ≤ α ≤ 1, if the proportion of natural numbers included as some aj is asymptotic to α. More formally, if we define the counting function A(x) as the... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers. ...

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 , does there necessarily exist n > n₀- such that n is refactorable and ?

History

First defined by Curtis Cooper and Robert E. Kennedy ^[2] where they showed that the tau numbers has 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[1]. Colton called such numbers "refactorable" While computer programs had discovered proofs before, this disovery 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. Dr. Curtis Cooper is a professor at the Central Missouri State University. ... In mathematics, a sequence a1, a2, ... , an, with the aj positive integers and aj < aj+1 for all j, has natural density α, where 0 ≤ α ≤ 1, if the proportion of natural numbers included as some aj is asymptotic to α. More formally, if we define the counting function A(x) as the... To meet Wikipedias quality standards, this article or section may require cleanup. ... A labeled graph with 6 vertices and 7 edges. ...

References

1. ^ J. Zelinsky, "Tau Numbers: A Partial Proof of a Conjecture and Other Results," Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8
2. ^ 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