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A process of successively crossing out members of a list according to a set of rules such that only some remain. The best known sieve is the Eratosthenes Sieve for generating Prime Numbers. In fact, numbers generated by sieves seem to share a surprisingly large number of properties with the Prime Numbers.
See also Happy Number, Number Field Sieve Factorization Method, Prime Number, Quadratic Sieve Factorization Method, Sierpinski Sieve, Sieve of Eratosthenes, Wallis Sieve
References
Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974.
Pomerance, C. ``A Tale of Two Sieves.'' Not. Amer. Math. Soc. 43, 1473-1485, 1996.