elementary number theory - Do such unique representations of positive integers exist?

It is well known that every positive integer $n>0$ can be represented uniquely in the form
$$n=2^k(2m+1),$$
for positive integers $k,m\geq0$. Does there exist one or more constants $c>1$ such that
$$2^k(2m+c)$$
is a unique representation for positive integers greater than some lower bound?

Answer

For $c=3$ the expression is unique, though you can not get a lot of numbers this way. For example, no power of $2$ can be written this way.

To see that, for the numbers which can be expressed this way, the expression is unique: Just remark that $$2^{k_1}(2m_1+3)=2^{k_2}(2m_2+3)\implies k_1=k_2\implies 2m_1+3=2m_2+3\implies m_1=m_2$$

A similar argument goes through for any odd $c$. For even $c$ the argument fails since we can not conclude that $k_1=k_2$.