def sum_of_divisors(n, sum_of_divisors_cache):
if n in sum_of_divisors_cache:
return sum_of_divisors_cache[n]
else:
sum_of_divisors = sum(all_proper_divisors(n))
sum_of_divisors_cache[n] = sum_of_divisors
return sum_of_divisors
def _abundant_numbers(less_than):
result = set()
for n in range(1, less_than):
if n < sum(all_proper_divisors(n)):
result |= {n}
return result