jmc

The hint is useful because it tells us that $2^{20}$ is equal to $1024^2$, which is slightly more than $1,000,000$, but is clearly less than $2,000,000$. Therefore, the largest power of $2$ which is less than $1,000,000$ is $2^{19}$. This tells us that $20$ of the integers smaller than $1,000,000$ are powers of $2$: $$2^0,2^1,2^2,2^3,\dots ,2^{17},2^{18},2^{19}.$$

However, we have to exclude the $7$ numbers $$2^0,2^3,2^6,2^9,2^{12},2^{15},2^{18}$$ from our count, because these are all powers of $8$ (in general, $2^{3n}$ is the same as $(2^3{)}^n$, which is $8^n$). That leaves us with $20-7=\boxed{13}$ powers of $2$ that aren't powers of $8$.