#brute solution

squares = []

for i in range(2, 101):
    for j in range(2,101):
        squares.append(i**j)
print len(list(set(squares)))

9183

print squares[:10]

[4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048]

def sum5digits(n):
    return sum([int(dig)**5 for dig in str(n)])

def sum4digits(n):
    return sum([int(dig)**4 for dig in str(n)])

cands1 = []

for i in range(1000,9999):
    if sum4digits(i) == i:
        cands1.append(i)

print cands1

[1634, 8208, 9474]

def solve30():
    cands = []
    for i in range(2,355000):
        if sum5digits(i) == i:
            cands.append(i)
    return sum(cands)

print solve30()

443839