ASYMPTOTIC SEMISMOOTHNESS PROBABILITIES

lWe call an integer semismooth with respect to y and z if each of its prime factors is less than or equal to y, and all but one are less tha n or equal to z. Such numbers are useful in various factoring algorith ms, including the quadratic sieve. Let G(alpha, beta) be the asymptoti c probability that a random integer n is semismooth with respect to n( beta) and n(alpha). We present new recurrence relations for G and rela ted functions. We then give numerical methods for computing G, tables of G, and estimates for the error incurred by this asymptotic approxim ation.