An integer is 
-smooth if it has no prime factors 
. The following table gives the first
few 
-smooth numbers for small 
. Berndt (1994, p. 52) called the 7-smooth numbers "highly
composite numbers."
 | OEIS |  -smooth numbers |
| 2 | A000079 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... |
| 3 | A003586 | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ... |
| 5 | A051037 | 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, ... |
| 7 | A002473 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, ... |
| 11 | A051038 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, ... |
The probability that a random positive integer 
is 
-smooth is 
, where 
is the number of 
-smooth numbers 
. This fact is important in application of Kraitchik's
extension of Fermat's factorization method
because it is related to the number of random numbers which must be examined to find
a suitable subset whose product is a square.
Since about 
-smooth numbers must be found (where 
is the prime
counting function), the number of random numbers which must be examined is about
. But because it takes
about 
steps to determine if a number is 
-smooth using trial division,
the expected number of steps needed to find a subset of numbers whose product is
a square is ![∼[pi(k)]^2n/psi(n,k)](/images/equations/SmoothNumber/Inline19.svg)
(Pomerance 1996). Canfield et al. (1983) showed that this function is minimized
when
 |
(1)
|
and that the minimum value is about
 |
(2)
|
In the continued fraction factorization algorithm, 
can be taken as 
, but in Fermat's
factorization method, it is 
. 
is an estimate for the largest prime
in the factor base (Pomerance 1996).
The curiosity
 |
(3)
|
involves the largest consecutive 19-smooth numbers, 11859210 and 11859211.
