An integer n is (y,z)-semismooth if n=pm where m is an integer with
all prime divisors ≤y and p is 1 or a prime ≤z. arge quantities of
semismooth integers are utilized in modern integer factoring algorithms, such
as the number field sieve, that incorporate the so-called large prime variant.
Thus, it is useful for factoring practitioners to be able to estimate the value
of Ψ(x,y,z), the number of (y,z)-semismooth integers up to x, so that
they can better set algorithm parameters and minimize running times, which
could be weeks or months on a cluster supercomputer. In this paper, we explore
several algorithms to approximate Ψ(x,y,z) using a generalization of
Buchstab's identity with numeric integration.Comment: To appear in ISSAC 2013, Boston M