Perfect Number Calculator

This perfect number calculator checks whether a whole number is perfect, meaning its proper divisors, all the divisors except the number itself, add up to exactly the number. Enter a value and the calculator gives the sum of the proper divisors and classifies the number as perfect, deficient (the divisors fall short) or abundant (they overshoot). Perfect numbers are rare and beautiful: 6 and 28 are the first two, and they connect to Mersenne primes through a result of Euclid and Euler. The definition, a worked example and the assumptions are set out below.

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Whole number (2 to 1,000,000,000)

Sum of proper divisors

Classification

The formula

Add up all the proper divisors of n (every divisor except n itself). If the sum equals n the number is perfect; if it is more, n is abundant; if less, n is deficient.

Worked example

The proper divisors of 28 are 1, 2, 4, 7 and 14, which add to 28, so 28 is perfect. The number 12 has proper divisors adding to 16, so it is abundant. Enter 28 to confirm.

Frequently asked questions

What is a perfect number?

A number equal to the sum of its proper divisors, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.

What are deficient and abundant numbers?

Deficient means the proper divisors add up to less than the number, abundant means they add up to more.

Are there odd perfect numbers?

None has ever been found, and whether any exist is a famous open question. All known perfect numbers are even.

Who this calculator is for

This calculator is for students, programmers and anyone exploring number theory.

What this calculator assumes

You enter a whole number within the stated range.
The exact integer algorithm is used.
Very large results are shown grouped for readability.

Formula and sources

Related calculators

Prime Factorization Calculator: break a number into prime factors.
LCM Calculator: lowest common multiple.
GCF Calculator: greatest common factor.

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