Powersmooth Number

A positive integer is -powersmooth if, whenever is a prime number dividing and divides but does not, the prime power satisfies . Thus is 5-smooth, but it is not 5-powersmooth because the maximal prime powers dividing it are 16, 9, and 5. Since all three are at most 16, the integer 720 is 16-powersmooth.

The -powersmooth numbers are exactly the positive divisors of

Consequently, there are only finitely many -powersmooth numbers for each fixed , unlike the -smooth numbers.