In looking at OEIS sequence A063539, $1,8,12,16,18,24,27,30,32,36,40,45,...$ I noticed that the first 1000 members were less than 4000, and thought there were no large gaps between them. What (if any) is the limiting density of this set of $\sqrt {n-1}$-smooth numbers? Is it more than 1/4? More importantly, is there a finite bound on the gap size? Specifically, let $d_i= a_{i+1} - a_i$ represent the size of the $i$th gap between consecutive members, is there a finite $C$ larger than all the $d_i$?

While I would appreciate references that answer this question, I would settle for some guiding intuition that suggests how to determine $C$ or even a very slowly growing $C(n)$ upper bound on the gaps. Note that for the density Dickman's function doesn't quite work as (for u=2) it includes all integers less than $\sqrt{n}$ and then some. If someone can show me that the error is small, I would accept using Dickman's function.

Gerhard "Hoping For A Royal Road" Paseman, 2016.11.21.

In looking at OEIS sequence A063539, $1,8,12,16,18,24,27,30,32,36,40,45,...$ I noticed that the first 1000 members were less than 4000, and thought there were no large gaps between them. What (if any) is the limiting density of this set of $\sqrt {n-1}$-smooth numbers? Is it more than 1/4? More importantly, is there a finite bound on the gap size? Specifically, let $d_i= a_{i+1} - a_i$ represent the size of the $i$th gap between consecutive members, is there a finite $C$ larger than all the $d_i$?

While I would appreciate references that answer this question, I would settle for some guiding intuition that suggests how to determine $C$ or even a very slowly growing $C(n)$ upper bound on the gaps. Note that for the density Dickman's function doesn't quite work as (for u=2) it includes all integers less than $\sqrt{n}$ and then some. If someone can show me that the error is small, I would accept using Dickman's function.

Gerhard "Hoping For A Royal Road" Paseman, 2016.11.21.