Prime power triples

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?

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter