Imagine we have some set of positive integers $\{k_1,k_2,...k_n\}$. We are given this formula

$$ x = \dfrac{\sum_{i=1}^{n}3^{n-i}2^{\sum_{j=1}^{i-1}k_j}}{2^{\sum_{i=1}^nk_i}-3^n} $$

and need to find if there will every be an integer solution for any set given besides the trivial solution $\{2\}$


Say the set was $\{2, 3, 1\}$, then our equation would be

$$ x = \dfrac{3^2+3*2^2+2^{2+3}}{2^{2+3+1}-3^3} = \dfrac{53}{37} $$

Even saying how to go about solving a problem like this with a hint would be incredibly helpful. Thank you!


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