# Density of integers $n$ with all prime factors of order $O(\log n)$?

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.

• 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? Feb 16, 2019 at 2:31
• @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?
– gen
Feb 16, 2019 at 9:13
• 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. Feb 27, 2019 at 14:44

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:

https://dms.umontreal.ca/~andrew/PDF/msrire.pdf

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

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