I read about a sequence similar to this one here on Stack Exchange a while back, somebody used it as an example for something that I can't recall! However, when I read about it it made me come up with this one, so here we are.
Let $\beta_n$ be the greatest number of the form $1/k$, with $k$ an integer, such that $n | k$ and $\sum_{m=1}^n\beta_m \lt 1.$ i.e. $\beta_1=1/2, \beta_2=1/4, \beta_3=1/6$, etc.
I would like to put some bounds on the sequence $k_n=1/\beta_n$, or possibly even some kind of asymptotic relation. However, the sequence soon begins growing extremely rapidly, with $k_{10}$ in the neighborhood of $10^{36}$, and this rapid increase makes it a bit harder to get accurate estimates on $k$. Could anyone here point me in the right direction for how to do that? I'm especially interested in an upper bound on the terms. I'm not really looking for a full solution, just a few ideas for me to push off of.
Thanks in advance everybody!