Perfect Number Checker – Divisor Sums

A perfect number equals the sum of its proper divisors (all divisors except itself). For 6: the divisors 1, 2, 3 sum to exactly 6. For 28: 1, 2, 4, 7, 14 sum to 28. The checker lists every proper divisor with a running total, then classifies the result — sum less than n is deficient (8 → 1+2+4 = 7), sum greater is abundant (12 → 1+2+3+4+6 = 16). Perfect numbers are vanishingly rare: 6, 28, 496, 8128, then a leap past 33 million. All known examples are even and follow Euclid’s form 2^(p−1)(2^p − 1) with a Mersenne prime: p = 2 gives 6, p = 3 gives 28, p = 5 gives 496, and p = 7 gives 8128. Whether any odd perfect number exists is an open problem.

1. 1

   Enter a number

   Any positive integer, e.g. 28

2. 2

   List proper divisors

   All divisors except n itself, with a running sum

3. 3

   Compare with n

   Equal → perfect; below → deficient; above → abundant

4. 4

   Note the Euclid form

   Even perfects match 2^(p−1)(2^p−1) for Mersenne primes p

• 1

  Perfect ⇔ proper divisor sum equals the number itself

• 2

  First four: 6, 28, 496, 8128 — then a huge gap

• 3

  σ(n) = 2n is the compact test (σ includes n)

• Including n itself among "proper" divisors

• Calling 12 perfect — its proper divisors sum to 16, making it abundant

• Assuming odd perfects exist; none are known

• Forgetting 1 counts as a proper divisor