Growth of the set $\lbrace 2^n+3^m \rbrace$ in the integers Let $A=\lbrace 2^n+3^m \ | \ n,m\in{\mathbb N}\rbrace$, and denote by $a_k$ the $k$-th element of $A$ in order from smallest to greatest ; thus $a_1=2,a_2=3,a_3=4,a_4=5,a_5=7\ldots$. Is it known whether $\frac{a_{k+1}}{a_k}$ has a limit when $k\to\infty$ ?
Weaker question : is it true that $\frac{a_{k+1}}{a_k} \leq \frac{3}{2}$ for every $k$ ? I have checked this for $k\leq 1500$.
My thoughts : if we write $a_k=2^n+3^m$, then $a_k<2^{n+1}+3^m$ so $a_{k+1} \leq 2^{n+1}+3^m \leq 2a_k$, so if the limit exists it is $\leq 2$. Numerical evidence suggests that the $\liminf$ is $1$.

See, how many numbers $a_k$ are there below some huge $N$? At least this includes all of those for which $2^n\le{N\over2}$ and $3^m\le{N\over2}$ (and probably more), which is already $O(\log^2N)$, which pretty much forbids the existence of a limit > 1.
