An integer n is y-smooth if and only if (iff) all prime factors of n are ≤ y. An integer n is (y,z)-semismooth iff all prime factors of n are ≤ y, except possibly for one prime factor p which only has to satisfy p ≤ z.

The paper deals with the problem of estimating for given x, y, z the number Ψ(x, y, z) of (y, z)-semismooth numbers n ≤ x. As is pointed out, this task plays an important role in optimizing factorization algorithms.

Five different algorithms are presented that strongly depend on corresponding counting algorithms for y-smooth numbers. A comparison is based primarily on empirical results. As may be expected, there is a tradeoff between speed and accuracy. On balance, one algorithm is recommended; it presumes and uses the Riemann hypothesis and combines high accuracy with reasonable speed.