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Problem 87

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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:

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

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

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素数幂三元组

最小的、可以表示为一个素数的平方、一个素数的立方和一个素数的四次方之和的数是$28$。实际上，在小于$50$的数中，一共有$4$个数满足这一性质：

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

在小于五千万的数中，有多少个数满足上述性质？

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