A 
-smooth number is a positive
integer whose prime factors are all at most 
. The number 1 is included since 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, ... |
For a positive integer cutoff 
, a positive integer sampled
uniformly from 1, 2, ..., 
is 
-smooth with probability 
, where 
counts the 
-smooth numbers at most 
. 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.
In a heuristic model with independent uniform samples, about 
-smooth numbers must be found (where 
is the prime counting
function), so about 
samples must be examined. Because it takes about 
steps to determine whether a number is 
-smooth using trial division,
the estimated number of steps to obtain a subset whose
product is a square number
is approximately ![[pi(k)]^2X/psi(X,k)](/images/equations/SmoothNumber/Inline20.svg)
(Pomerance 1996, p. 1477). Using smooth-number estimates of Canfield et al.
(1983), the leading exponential scale for 
is optimized when
 |
(1)
|
as 
, with the estimated minimum
having leading exponential scale
 |
(2)
|
For factoring an integer 
, the auxiliary cutoff is about 
in the continued
fraction factorization algorithm, while Kraitchik's quadratic polynomial
uses an estimate 
.
Here 
estimates the largest prime in the factor
base (Pomerance 1996, p. 1477).
The curiosity
 |
(3)
|
involves the largest consecutive 19-smooth numbers, 11859210 and 11859211.
