Solution:

A number cannot be written in the given form if and only if it is a power of a prime. We can see this by considering the prime factorization. Suppose that $k = p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{n}^{e_{n}}$, with $p_{1}, \ldots, p_{n}$ primes. Then we can write $m = p_{1}^{e_{1}}$ and $n = p_{2}^{e_{2}} \cdots p_{n}^{e_{n}}$. So, we want to find the powers of primes that are less than or equal to $100$.

There are $25$ primes, as given in the problem statement. The squares of primes are $2^{2}, 3^{2}, 5^{2}, 7^{2}$. The cubes of primes are $2^{3}, 3^{3}$. The fourth powers of primes are $2^{4}, 3^{4}$. The fifth powers of primes are $2^{5}$. The sixth powers of primes are $2^{6}$. There are no seventh or higher powers of primes between $2$ and $100$. This adds $10$ non-primes to the list, so that in total there are $10 + 25 = 35$ such integers.