Digit fifth powers

Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:

\[ \begin{align} 1634 = 1^4 + 6^4 + 3^4 + 4^4\\ 8208 = 8^4 + 2^4 + 0^4 + 8^4\\ 9474 = 9^4 + 4^4 + 7^4 + 4^4 \end{align} \]

As \( 1=1^{4} \) is not a sum it is not included.

The sum of these numbers is \( 1634+8208+9474=19316 \).

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.