pyPerfectNumbers

# Find perfect Numbers based on the Euclid-Euler Theorem
# Mike Kerry - Sept 2022
#
# Consider the sequence 1, 2, 4, 8, 16 ....
# If the sum of this sequence is prime, then
# the product of the sum and the last in the sequence is perfect


def is_prime(num):
    if num < 4:
        return num > 1  # 2 and 3 => True, anything less than 2 is False
    if not num % 2:
        return False    # All even numbers above 2, are non-prime
    for d in range(3, 1 + int(num ** 0.5), 2):
        if not num % d:
            return False
    return True


ctr, tot = 0, 1
n = 1
while ctr < 8:
    n += n               # double-up n each iteration
    tot += n             # sum of the sequence 1, 2, 4, 8, ...
    if is_prime(tot):    # if this sum is prime ...
        print(n * tot)   # ... then (sum * last) is 'perfect'
        ctr += 1

# Output
# 6
# 28
# 496
# 8128
# 33550336
# 8589869056
# 137438691328
# 2305843008139952128