# 3-smooth number

Let $(u_n)_{n\in\mathbb N}$ be the sequence of $3$-smooth numbers (that is whole number that can be written as $2^a3^b$ with $a,b\in\mathbb N$) sorted by increasing order. I am looking for the principal term of the asymptotic of $\log(u_n)$ as well as the error term, for $n\to+\infty$. Any idea to achieve that?

• Closely related: the number of smooth numbers up to a given bound: math.stackexchange.com/questions/1182775/… – Wojowu Jun 25 '18 at 11:44
• Related, yes, but does it give an answer to my problem? – joaopa Jun 25 '18 at 11:50
• You can deduce from there $\log u_n\sim\sqrt{2\log 2\log 3}\sqrt{n}$. If you figure out an error term for the number of smooth numbers, you can translate it to an error term to your problem. – Wojowu Jun 25 '18 at 12:22
• How do you obtain this equivalent? I can not s the link between the number of $u_k$ lesser than $n$ and $u_n$. – joaopa Jun 25 '18 at 12:27
• @joaopa: Let $N(x)$ denote the number of $u_n$'s up to $x$. Then, according to the link above, $N(x)\sim(\log x)^2/(2\log 2\log 3)$. Hence $n=N(u_n)\sim(\log u_n)^2/(2\log 2\log 3)$, which gives $\log u_n\sim\sqrt{(2\log 2\log 3)n}$. – GH from MO Jun 25 '18 at 18:05

3-smooth numbers are tabulated at http://oeis.org/A003586 and there you will also find the asymptotic, $${1\over\sqrt6}e^{\sqrt{2(\log2\log3)n}}$$ (attributed to Benoit Cloitre) as well as several links to the literature.