2nd Update: The paper was published in the INTEGERS Journal, it is the 33rd paper in this list: math.colgate.edu/~integers/vol22.html

Update: my friend and I did some further research in this direction and we wrote a paper on this topic: researchgate.net/publication/…

oeis.org/A340163 here's a link to the sequence of $A_k$ values.

I've checked manually that the smallest values for $A_k$, starting from $k=1$, are $$(A_k)=(0,0,1,2,3,7,14,28,57,115,233,469,...)$$ which is really close to the one I derived: $$(A_k)=(0,1,1,2,4,8,16,32,64,128,256,512,...)$$

@user142929 Also, I'm really close to proving another generalization conjecture related to the (ii) Ramanujan Floor problem and I think I might have a better value for $A_k$ in this conjecture. I'll let you know about the progress!