Prove $\prod_{i=1}^n \frac{3K_i+2}{2K_i+1}$ can never be a power of $2$
Its not true. For instance, define $\mathcal K:=\{5,6,8,24,27,32,41,47,69,92\}$. Then we have, $$\prod_{i\in\mathcal K}\frac{3i+2}{2i+1}=2^6.$$
Prove $\prod_{i=1}^n \frac{3K_i+2}{2K_i+1}$ can never be a power of $2$
Its not true. For instance, define $\mathcal K:=\{5,6,8,24,27,32,41,47,69,92\}$. Then we have, $$\prod_{i\in\mathcal K}\frac{3i+2}{2i+1}=2^6.$$