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."

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
(Pomerance 1996). Canfield et al. (1983) showed that this function is minimized
when

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.Blecksmith,
R.; McCallum, M.; and Selfridge, J. L. "3-Smooth Representations of Integers."
Amer. Math. Monthly105, 529-543, 1998.Canfield, E. R.;
Erdős, P.; and Pomerance, C. "On a Problem of Oppenheim Concerning 'Factorisation
Numerorum.' " J. Number Th.17, 1-28, 1983.Mintz,
D. J. "2, 3 Sequence as a Binary Mixture." Fib. Quart.19,
351-360, 1981.Pomerance, C. "On the Role of Smooth Numbers in Number
Theoretic Algorithms." In Proc. Internat. Congr. Math., Zürich, Switzerland,
1994, Vol. 1 (Ed. S. D. Chatterji). Basel: Birkhäuser, pp. 411-422,
1995.Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc.43, 1473-1485, 1996.Ramanujan, S. Collected
Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar,
and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. xxiv, 2000.Sloane,
N. J. A. Sequences A000079/M1129,
A002473/M0477, A003586,
A051037, and A051038
in "The On-Line Encyclopedia of Integer Sequences."