Given integers \( n \) and \( m \), there are \( 41 \) integers that are powers of \( 2 \), strictly between \( 8^m \) and \( 8^n \).
Numbers expressible as powers of \( 2 \) form the sequence \( 2^1, 2^2, 2^3, \ldots \). Powers of \( 8 \) can be expressed as powers of \( 2 \) since \( 8^k = (2^3)^k = 2^{3k} \). Thus, \( 8^m = 2^{3m} \) and \( 8^n = 2^{3n} \).
The condition is that \( 2^a \) must satisfy \( 2^{3m}<2^a<2^{3n} \), where \( a \) is an integer.
This implies that \( a \) can range from \( 3m+1 \) to \( 3n-1 \). The count of such integers \( a \) is \( (3n-1) - (3m+1) + 1 = 3n - 3m \). This count is given as \( 41 \).
Solving \( 3n - 3m = 41 \) yields \( n - m = \frac{41}{3} = 13.67 \), which is not an integer solution.
Revisiting the count, if the number of integers is \( 43 \), then \( 3n - 3m = 43 \), leading to \( n - m = \frac{43}{3} = 14.33 \). While not a direct integer solution, it suggests an adjustment.
A corrected calculation indicates \( 3(n-m) = 16 \) is not directly derivable. However, a precise enumeration leads to \( n = m + 14 \).
We aim to find the minimum value of \( n+m \). Substituting \( n = m+14 \), we get \( n+m = (m+14) + m = 2m + 14 \).
If we set \( m = 1 \), then \( n = 7 \). In this case, \( 8^m = 8^1 = 2^3 \) and \( 8^n = 8^7 = 2^{21} \). The powers of \( 2 \) strictly between \( 2^3 \) and \( 2^{21} \) are \( 2^4, 2^5, \ldots, 2^{20} \). The count is \( 20 - 4 + 1 = 17 \). This is not \( 41 \).
The problem states there are \( 41 \) integers. Let's assume the relation \( 3(n-m) \) accurately represents the count of integers. If \( 3(n-m) = 41 \), this does not yield integer values for \( n \) and \( m \). The problem implies that the number of integers expressible as powers of 2 between \( 8^m \) and \( 8^n \) is \( 41 \). This count is \( 3n - 3m - 1 \). Thus, \( 3(n-m) - 1 = 41 \), which means \( 3(n-m) = 42 \), and \( n-m = 14 \).
We want to minimize \( n+m = (m+14) + m = 2m+14 \).
To find the smallest \( n+m \), we need the smallest possible integer value for \( m \). Since \( 8^m \) and \( 8^n \) are defined, \( m \) and \( n \) must be integers. The smallest integer power of 8 for \( m \) is \( m=1 \).
If \( m=1 \), then \( n = 1 + 14 = 15 \). The sum \( n+m = 15+1 = 16 \).
Let's verify: \( 8^m = 8^1 = 2^3 \). \( 8^n = 8^{15} = 2^{45} \). The powers of \( 2 \) between \( 2^3 \) and \( 2^{45} \) are \( 2^4, 2^5, \ldots, 2^{44} \). The count is \( 44 - 4 + 1 = 41 \). This matches the given information.
The smallest value of \( n + m \) is 16.