I am trying to find some restrictions for what the exponents, $a_{i}\in\mathbb{Z}^{+}$, of two powers can be if the following equation yields an odd integer:
$[3^{n}+3^{(n-1)}2^{(a_{2})}+3^{(n-2)}2^{(a_{2}+a_{3})}+...+3^{(1)}2^{(a_{2}+..+a_{n-1})}+3^{(0)}2^{(a_{2}+..+a_{n-1}+a_{n})}]/[2^{(a_{1}+a_{2}+...+a_{n})}-3^{n}] = m$
Where $n\geq1$, $m\in2\mathbb{Z}+1$, and $a_{i}\in\mathbb{Z}^{+}, a_{i}>0$
What divisibility rules can I use to place restrictions on what values $a_{i}$ can take?
Written in another form: $3^{n}+3^{(n-1)}2^{(a_{2})}+3^{(n-2)}2^{(a_{2}+a_{3})}+...+3^{(1)}2^{(a_{2}+..+a_{n-1})}+3^{(0)}2^{(a_{2}+..+a_{n-1}+a_{n})}=m(2^{(a_{1}+a_{2}+...+a_{n})}-3^{n})$