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The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} 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 $\{n\le x:p_1(n)<y\}$?

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 $p_1$.

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    $\begingroup$ Look up the fundamental lemma of sieve theory, it answers your question and more. $\endgroup$ Dec 9, 2023 at 13:04
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    $\begingroup$ 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. $\endgroup$ Dec 9, 2023 at 17:29
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    $\begingroup$ 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...). $\endgroup$ Dec 9, 2023 at 17:33

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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).

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