# Is there a way to estimate the number of positive integers less than or equal to $n$ that have a given prime $p$ as a least prime factor

The probability that an integer $p$ divides an integer $x$ is $\dfrac{1}{p}$.

From this article on almost prime numbers, the number $\pi_k(n)$ of positive integers less than or equal to $n$ with at most $k$ divisors (not necessarily distinct) is asymptotic to:

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

Is there a way to estimate the number of positive integers less than or equal to $n$ that have a given prime $p$ as the least prime factor.

As an example, the proportion of numbers that have $7$ as least prime factor is about $\frac 12 \cdot \frac 23 \cdot \frac 45 \cdot \frac 17=\frac 8{210}$ This will be exact for $n$ a multiple of $210$.

More generally the fraction is $$\frac1p\prod_{\substack {q \lt p \\q \text{ prime}}}\frac {q-1}q$$ which will be exact if $n$ is divisible by $p$ primorial. The numerator is given in A005867, but no asymptotic value is given. The denominator is given in A002110, with the value up through the $k$th prime being asymptotic to $k^{k(1+o(1))}$