That is, I have that smallest number not representable by $d=\{1,2,4,8,\dots,2^{n-2},2^{n-1}\}$ is $f_d(n)+1$ where $f_d(n)=2^n-1+g(k)$.
The Main Question is: Can we find $d$ such that we can show $f_d(n)$ grows even faster than the above one for $\{1,2,4,8,\dots,2^{n-2},2^{n-1}\}$ ?
Regarding improving the lower bound on my newest problem.