Sorry for the repost and for my "bad" English. I made a lot of errors in the previous one, so here's my actual question:
Let's take a look at this sequence:
(1)
$[a_1,a_2,a_3,a_4,...,a_x]$
where
$$a_1=\frac{3 \cdot 2^{n_1}+2^{n_2}}{2^{n_3}-9}$$
$$a_2=\frac{9 \cdot 2^{n_1}+3 \cdot 2^{n_2}+2^{n_3}}{2^{n_4}-27}$$
$$a_x=\frac{3^{x}\cdot 2^{n_1}+3^{x-1}\cdot 2^{n_2}+3^{x-2}\cdot 2^{n_3}+ ... +3 \cdot 2^{n_{x}}+2^{n_{x+1}}}{2^{n_{x+2}}-3^{x+1}}$$
$n_1,n_2,n_3,...,n_x ∈ \mathbb N$
$n_1≤n_2≤n_3≤...≤n_x$
Is there a way to prove that every term of (1) is not a natural number?
Thanks.