Comparing $\pi(x)$ and $\pi^{(k)}(x)$

Question

We say a k-almost prime is an integer that results as the product of k prime, counting repetition. For example, $12$ is a $3$-almost prime as $12= 3 \times 2 \times 2$. Additionally, we define $\pi^{(k)}(n)$ as the number of $k$-primes less than or equal to $n$. The standard prime counting function $\pi(n)$ is therefore referred to as $\pi^{(1)}(n)$. I asked myself the following question:

For every $k\geq2$, does there exists a corresponding $c$ such that $$\pi^{(1)}(n)<\pi^{(k)}(n)$$ is true for all $n\geq c$. If so, how could one find the corresponding $c$?

In other words, does there exists a $c$ for every $k$ after which there will be more $k$-primes than primes? At first, it seemed obvious that this was true for every k. However, after thinking more deeply about the question, I realized that:

1. Prime k-tuples may suddenly increase the number of primes and surpass the number of $k$-primes at some $n$.

2. Multiplying $k$ primes together will created very large numbers, meaning that they will only be counted in $\pi^{(k)}(n)$ for some equaly large $n$.

3. Choosing a very large k means that there will be no $k$-primes less than $\prod_{n=1}^{k} p(n)$, where $p(n)$ is the $n^{th}$ prime.

There exists an equation for $\pi^{(k)}(n)$ (found here), but it takes the sum over primes, meaning it is hard to make use of it in this case due to the unpredictable gaps between primes. Can the question above be answered? If so, what is the answer?

Answer 1

According to Landau,

$$\pi^{(k)}(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}\;.$$

This answers your question whether there is such a $c$ in the affirmative, but doesn't yet address your other question how to find one.

(In a previous version of this answer, I quoted this result from Wikipedia, but it seeems that it's misrepresented there, as it's taken to apply to $\sum_{j=1}^k\pi^{(j)}$ instead.)

Edit: I found Landau's Handbuch der Lehre von der Verteilung der Primzahlen ("Handbook of the Theory of the Distribution of the Prime Numbers"), from which this was quoted, at archive.org. The proof of the above result starts on p. $205$; the result itself is equation $(5)$ on p. $211$ (with slightly different notation). If you don't understand German, I'll be happy to help.