Digit fifth powers

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

1634 = 1⁴ + 6⁴ + 3⁴ + 4⁴
8208 = 8⁴ + 2⁴ + 0⁴ + 8⁴
9474 = 9⁴ + 4⁴ + 7⁴ + 4⁴

As 1 = 1⁴ 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.

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Idea

Naive solution. The key is the upper bound.

It is always true that $ n * 9^k \lt 10^{n+1}, if \; k \lt 10 $.

So the upper bound for numbers that can be written as the sum of fifth powers of their digits is $ 10^{5+1} $

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def digits_power_sum(n, power):
    s = 0
    while n:
        s += pow(n % 10, power)
        n //= 10
    return s

digits_power_sum(1634, 4)

1634

def solve(power):
    upper_bound = pow(10, (power+1))
    s = 0
    for i in range(2, upper_bound):
        if i == digits_power_sum(i ,power):
            s += i
    return s

solve(4)

19316

solve(5)

443839