1. First you compute the density of the set $F_r$ of numbers having exactly $r$ prime factors equal to $2$ mod $3$ and NO other factors. The computation should be completely parallel to the computation of density of $r$-almost primes, which is discussed at length, for example, in this question. (or google for more references). call the resulting density $delta(r);$ \delta(r);$for primes the density is$1/\log x$2. Since you already know the result for$r=0$(f(x)$f(x) = x/(\log^{1/2} x))x)$), you simply compute the integral$\int_1^N \lfloor f(N/x)\rfloor d \delta(x).$(it seems clear that the result for$k$-almost primes in at least the original Landau form can be derived in exactly this fashion). 1 This is a sketch, but I would think the details are reasonably routine: 1. First you compute the density of the set$F_r$of numbers having exactly$r$prime factors equal to$2$mod$3$and NO other factors. The computation should be completely parallel to the computation of density of$r$-almost primes, which is discussed at length, for example, in this question. (or google for more references). call the resulting density$delta(r);$for primes the density is$1/\log x$2. Since you already know the result for$r=0$(f(x) = x/(\log^{1/2} x)), you simply compute the integral$\int_1^N \lfloor f(N/x)\rfloor d \delta(x).$(it seems clear that the result for$k\$-almost primes in at least the original Landau form can be derived in exactly this fashion).