A number
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 , where 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
such that
and
are both refactorable numbers are 1, 8, 1520, 50624, 62000, 103040, ... (OEIS A114617).
Zelinsky (2002) proved that there are no refactorable numbers and such that and also Colton's conjecture that no three consecutive
integers can all be refactorable.
Colton, S. "Refactorable Numbers--A Machine Invention." J. Integer Sequences2, 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 Scientist2150,
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 Sequences5,
No. 02.2.8, 2002. http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html.
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 
, where 
is the divisor function.
such that 
and 
are both refactorable numbers are 1, 8, 1520, 50624, 62000, 103040, ... (OEIS A114617).
and 
such that 
and also Colton's conjecture that no three consecutive
integers can all be refactorable.
