The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is $28$. In fact, there are exactly four numbers below fifty that can be expressed in such a way:

$$\begin{align} 28 &= 2^2 + 2^3 + 2^4\\ 33 &= 3^2 + 2^3 + 2^4\\ 49 &= 5^2 + 2^3 + 2^4\\ 47 &= 2^2 + 3^3 + 2^4 \end{align}$$

How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?