Prime power triples

[mattgathu]

Problem Number: 87

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:
28 = 22 + 23 + 24
33 = 32 + 23 + 24
49 = 52 + 23 + 24
47 = 22 + 33 + 24
How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?

Ref

https://projecteuler.net/problem=87