  In this paper we present and prove rapidly convergent formulas for the
distribution of the $3$-smooth, $5$-smooth, $7$-smooth and all other smooth
numbers. One of these formulas is another version of a formula due to Hardy and
Littlewood for the arithmetic function $N_{a,b}(x)$, which counts the number of
positive integers of the form $a^{p}b^{q}$ less than or equal to $x$.
