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Edit: I have a program running to find pairs $(2^k,3^l)$ for which the first two statements can be satisfied, and for each pair, search for a witness to $F(s)>1$. For $k\le 219$ and $m\le 25$, there are 13 pairs with no witness of the form $(4^n,q)$. The much more computationally intensive task of confirming absence of a witness of the form $(4^n p,q)$ is likely intractable if the pair is a solution. Witnesses have been found for the smallest 7 pairs, the first pair with no known witness yet is $(2^{141}, 3)$.

Since $s\equiv 2\pmod{3}$ is necessary, $s-4$ cannot be a multiple of 3. If we define $k_m(s)$ to be the smallest $k\ge 2$ such that $(k,s-4)=1$, then we can refine the first bullet point above:

Edit: I have a program running to find pairs $(2^k,3^l)$ for which the first two statements can be satisfied, and for each pair, search for a witness to $F(s)>1$. For $k\le 219$ and $m\le 25$, there are 13 pairs with no witness of the form $(4^n,q)$. The much more computationally intensive task of confirming absence of a witness of the form $(4^n p,q)$ is likely intractable if the pair is a solution. Witnesses have been found for the smallest 10 pairs, the first pair with no known witness yet is $(2^{187}, 3)$.

Since $s\equiv 2\pmod{3}$ is necessary, $s-4$ cannot be a multiple of 3. If we define $k_m(s)$ to be the smallest $k\ge 2$ such that $(4^k-1,s-4)=1$, then we can refine the first bullet point above:

I have checked the value of $F(s_{a,30})$ for $a \le 500$ and found 4 probable zeros and 7 probable solutions, the largest of which is $(4^{31}*459703, s_{34,30}-4^{31}*459703)$. This might be the largest solution.

I have checked the value of $F(s_{a,30})$ for $a \le 500$ and found 4 probable zeros and 7 probable solutions, the largest of which is $(4^{31}*459703, s_{34,30}-4^{31}*459703)$. This might be the largest solution. (Update: I did finish the check of the less-probable cases, completing a proof that this pair is a solution. The proof is found here).

We may define $g(k)$ to be the smallest value sharing a factor with $(4^n-1)/3$ for all $n\le k$, so $k_m(g(k))=k+1$, then $g(2)=5$, $g(3)=35$, $g(5)=385$, etc. Each defines an arithmetic progression with exceptionally low values of $F(s)$. Specifically,

We may define $g(k)$ to be the smallest value not divisible by 3 and sharing a factor with $4^n-1$ for all $n\le k$, so $k_m(g(k))=k+1$, then $g(2)=5$, $g(3)=35$, $g(5)=385$, etc. Each defines an arithmetic progression with exceptionally low values of $F(s)$. Specifically,