For a rational integer $n \in \mathbb{Z}_{+}$, let $\mathfrak{p}(n)$ denote the set of (distinct) prime factors of $n$. Then for a positive constant $c$, let $$f(x) = \vert\{n\in\mathbb{Z}_{+}:\ n\leq x,\textit{ and } \max{\mathfrak{p}(n)} \leq c \log n\}\vert.$$ I.e. $f(x)$ is the number of integers $n$ below $x$ such that there largest prime factor is bounded by $c\log n$. Is the density of such numbers $f(x)/x$ known? Could you refer me to some relevant results on this?

I am aware of some density results on B-smooth numbers but that's not what I'm looking for.

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    $\begingroup$ You are asking about smooth numbers, but at the same time, that's not you are looking for. Then what kind of answer are you looking for? $\endgroup$ Feb 16, 2019 at 2:31
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    $\begingroup$ @i707107 for smooth numbers the bound on the size of prime divisors is constant, for me its a variable. Let me know if I misunderstand something? $\endgroup$
    – gen
    Feb 16, 2019 at 9:13
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    $\begingroup$ OEIS has this sequence when $c = 1$ (oeis.org/A137845). I'd guess the density is zero from the data there. "Square root smooth" numbers (oeis.org/A048098 - replace $c \log n$ with $\sqrt{n}$ in your definition) have positive density. $\endgroup$ Feb 27, 2019 at 14:44

1 Answer 1


Let $f_u(x)$ denote the number of integers $n \le x$ that only have prime factors at most $x^{1/u}$. If $x$ is big eough, then $x^{1/u} \ge c \log(x) \ge c \log(n)$ for all $n \le x$. So, for big enough $x$,

$$f(x) \le f_u(x).$$

On the other hand, a therorem of Dickman from 1930 says that

$$\lim \frac{f_u(x)}{x} = \rho(u),$$

where $\rho(u)$ can be described fairly explicitly. For example, $\rho(u) = 1 - \log(u)$ for $1 \le u \le 2$. The explicit description of $\rho(u)$ also implies that $\lim_{u \rightarrow \infty} \rho(u) = 0$. In particular, your limit is also zero.

A good reference for these facts (and much more) is here:


(Everything above is contained in the first two pages of that survey, see (1.3) and (1.6))

  • $\begingroup$ For a derivation of the asymptotic to Dickman's function see this p.71 $\endgroup$
    – reuns
    Feb 28, 2019 at 1:24

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