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Refactorable Number


A number n is said to be refactorable, sometimes also called a tau number (Kennedy and Cooper 1990), if it is divisible by the number of its divisors sigma_0(n), where sigma_k(n) is the divisor function.

The first few refactorable numbers are 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, ... (OEIS A033950).

The first new n such that n and n+1 are both refactorable numbers are 1, 8, 1520, 50624, 62000, 103040, ... (OEIS A114617).

Zelinsky (2002) proved that there are no refactorable numbers a and b such that a-b=5 and also Colton's conjecture that no three consecutive integers can all be refactorable.


See also

Divisor Function

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References

Colton, S. "Refactorable Numbers--A Machine Invention." J. Integer Sequences 2, No. 99.1.2, 1999. http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html.Kennedy, R. E. and Cooper, C. N. "Tau Numbers, Natural Density, and Hardy and Wright's Theorem 437." Internat. J. Math. Math. Sci. 13, 383-386, 1990.Graham-Rowe, D. "Eureka!" New Scientist 2150, 17, Sep. 5, 1998.Sloane, N. J. A. Sequences A033950 and A114617 in "The On-Line Encyclopedia of Integer Sequences."Zelinsky, J. "Tau Numbers: A Partial Proof of a Conjecture and Other Results." J. Integer Sequences 5, No. 02.2.8, 2002. http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html.

Referenced on Wolfram|Alpha

Refactorable Number

Cite this as:

Weisstein, Eric W. "Refactorable Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RefactorableNumber.html

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