Refactorable Number

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.

REFERENCES:

Colton, S. "Refactorable Numbers--A Machine Invention." J. Integer Sequences 2, No. 99.1.2, 1999. https://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. https://www.cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html.