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).

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).

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).

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).