Perfect Number in Python

The question

Read a positive integer and print whether it is a perfect number: equal to the sum of its divisors other than itself (6 = 1 + 2 + 3).

The code

n = int(input())
total = 0
for d in range(1, n):
    if n % d == 0:
        total += d
if total == n:
    print(n, "is a perfect number")
else:
    print(n, "is not a perfect number")

How it works

1. 1Loop d from 1 to n − 1.
2. 2Whenever n % d == 0, d is a divisor — add it to total.
3. 3If total equals n, the number is perfect.

Dry run: is 28 a perfect number?

The loop tries every d from 1 to 27; only the values that divide 28 exactly (28 % d == 0) are added. Those are the rows below.

total = 28 = n, so 28 is a perfect number.

Faster: check divisors only up to n // 2

No number has a divisor bigger than its half (other than itself), so the loop can stop at n // 2. For 8128 that is half the work.

n = int(input())
total = 0
for d in range(1, n // 2 + 1):
    if n % d == 0:
        total += d
if total == n:
    print(n, "is a perfect number")
else:
    print(n, "is not a perfect number")

28

28 is a perfect number

Print all perfect numbers from 1 to 1000 using a function

Move the check into a function that returns True or False, then call it for every number in the range.

def is_perfect(n):
    total = 0
    for d in range(1, n):
        if n % d == 0:
            total += d
    return total == n

for n in range(1, 1001):
    if is_perfect(n):
        print(n)

(no input)

6
28
496

Questions students ask

What is a perfect number in Python?

A perfect number is equal to the sum of its divisors, leaving out the number itself. 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. In Python you find the divisors with n % d == 0 inside a loop and add them up.

What are the first four perfect numbers?

6, 28, 496 and 8128. The next one is 33,550,336, which is why exam questions never go past 8128.

Is 1 a perfect number?

No. 1 has no divisor other than itself, so the sum is 0, and 0 is not equal to 1. The program on this page prints "1 is not a perfect number".

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