This was not bad! the minimum number that can be written as a sum of its 5th digits powers is

$$ \begin{align*} 2 = 0^5 + 0^5 + 0^5 + 1^5 + 1^5. \end{align*} $$

while the maximum number that can be written as a sum of its 5th digits powers is

$$ \begin{align*} 295245 = 9^5 + 9^5 + 9^5 + 9^5 + 9^5. \end{align*} $$

So let’s check all the numbers in this range

// 9^5 + 9^5 + 9^5 + 9^5 + 9^5 = 5*9^5 = 295,245
// 0^5 + 0^5 + 0^5 + 1^5 + 1^5 = 2
int total_sum = 0;
for (int n = 2; n <= 295245; n++) {
    long sum = 0;
    int num = n;
    while (num > 0 && sum <= n) {
        int d = num%10;
        int p = pow(d, 5);
        sum += p;
        num /= 10;
    }
    if (sum == n) {
        total_sum += sum;
    }
}
printf("%d\n", total_sum);



References

Project Euler - 30