Powers of 2 between Powers of 8

We have $8^m = 2^{3m}$ and $8^n = 2^{3n}$.

The integers between $2^{3m}$ and $2^{3n}$ that are powers of 2 are: $2^{3m+1}, 2^{3m+2}, \dots, 2^{3n-1}$

The total number of such terms is: $(3n - 1) - (3m + 1) + 1 = 3n - 3m - 1$

Given that there are 41 such integers: $3n - 3m - 1 = 41 \implies 3(n - m) = 42 \implies n - m = 14$

Since $n$ and $m$ are positive integers, the minimum value for $m$ is $1$. Then $n = m + 14 = 15$.

Thus, the smallest possible value of $n + m = 15 + 1 = 16$.