rowlands sequence is defined as follows
\begin{equation} a_{n}=a_{n-1} + b_{n} \end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural Prime-Generating Recurrence".
what upper and lower bounds exists for $a_{n}$?
i managed to prove $a_{n} < Cn$ for a constant $C$ depending on $h$ by substituting the $gcd-lcm$ identity into the above equation and then substituting that equation into itself into getting the following and bounding it from above
$a_{n} = a_{h}\prod_{i=h+1}^{n}\left(1+\frac{i}{\operatorname{lcm}\left(a_{i-1},\ i\right)}\right)$
but from numerical data it seems $\frac{3}2(n+1)\le a_{n} \le 3n$ are the best bounds for any initial condition where $a_{h}>h$.
remark: showing that bound is equivelent to showing
$ \frac{3}2h\prod_{i=h}^{n}\left(1+\frac{1}{i}\right)\le a_{h}\prod_{i=h+1}^{n}\left(1+\frac{i}{\operatorname{lcm}\left(a_{i-1},\ i\right)}\right)\le3h\prod_{i=h}^{n-1}\left(1+\frac{1}{i}\right) $
what potential ways could the bounds be derived?
I've cross posted this on math stackexchange
