# Smallest prime factor of numbers

The literature refers to smooth integers as $$$$\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},$$$$ where $$P_1(n)$$ is the largest prime factor of $$n$$. There are lots of results studying $$\Psi(x,y)$$, but I am interested in a least prime factors analogue. Is there a function, similar to $$\Psi$$, which counts $$\{n\le x:p_1(n)\ge y\}$$ or $$\{n\le x:p_1(n)?

I want to find a good estimate for $$\#\{n\le x:p_1(n)\le m\}$$; such an estimate exists for $$P_1$$ but I cannot find a similar one for $$p_1$$.

• Look up the fundamental lemma of sieve theory, it answers your question and more. Dec 9, 2023 at 13:04
• To elaborate on Stanley's comment and GH from MO's comment: an integer with $p_1(n)>y$ is called $y$-rough or $y$-sifted. Let $\Phi(x,y) := \#\{ n \le x: p_1(n)>y\}$. Set $u:=\log x / \log y$. There are 3 regimes: if $u \le 2$ then $\Phi(x,y)$ counts primes in $(y,x]$, so $\Phi(x,y) = \pi(x)-\pi(y) \sim x/\log x$ if $y=o(x)$. If $u \to \infty$ then $\Phi(x,y) \sim x\prod_{p \le y}(1-1/p) \asymp x/\log y$ by the fundamental lemma of sieve theory. If $u\ge 2$ is bounded (or does not grow too quickly) then $\Phi(x,y) \sim \omega(u) x / \log y$ for a function known $\omega$ as Buchstab function. Dec 9, 2023 at 17:29
• De Bruijn's 1950 paper "On the number of uncancelled elements in the sieve of Eratosthenes" is still a good source for these results, which can also be proved without sieve theory by utilizing Dirichlet series techniques. In any case, $\#\{ n \le x: p_1(n) \ge y\}$ is $o(x)$ once $y \to \infty$ and so $\#\{ n \le x: p_1(n) < y \} \sim x$ once $y \to \infty$ (for this, inclusion-exclusion suffices...). Dec 9, 2023 at 17:33

The number of integers $$1\leq n\leq x$$ with smallest prime factor exceeding $$y$$ is usually denoted by $$\Phi(x,y)$$. It has been studied thoroughly. See, for example, Chapter III. 6 (Integers free of small prime factors) in Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995).