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Perfect Number Checker: Perfect, Abundant, Deficient, Amicable

This free tool sums the proper divisors of a number and tells you if it is perfect, abundant or deficient. It also checks whether the number has an amicable partner and lists every divisor.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

Perfect numbers have fascinated mathematicians for more than 2,000 years. They are tied to Mersenne primes and to the oldest open questions in number theory.

Enter a number and the tool does the rest.

How to Use This Tool

  1. Enter or paste your input in the box or boxes above.
  2. Choose the options, if there are any.
  3. Press the button to run the tool.
  4. Read the result and use the copy button where one is shown.

How the Calculation Works

The tool finds all divisors by testing numbers up to the square root and pairing each with its partner. The proper divisor sum s(n) is the total without n itself. Compare s(n) with n to classify the number. For the amicable test it computes s(s(n)) and checks that it returns to n with s(n) different from n.

Worked Example

For 28 the proper divisors are 1, 2, 4, 7 and 14, which add to 28, so it is perfect. For 220 the sum is 284 and for 284 the sum is 220, so they are an amicable pair.

Common Uses

  • Number theory homework.
  • Programming puzzles.
  • Maths clubs and trivia.
  • Exploring divisor patterns.

Tips and Common Mistakes

  • Numbers above one trillion are not accepted.
  • The divisor list is shown only up to 5,000 divisors.
  • 1 is deficient because its proper divisor sum is 0.
  • Check the first result against a case you already know, so you can trust the tool for the rest.
  • Nothing you type leaves your device. The work is done by your browser, so it also works with a slow connection once the page has loaded.

Next, you may need the Collatz sequence calculator, triangular number calculator, prime number checker, GCD and LCM calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

A number equal to the sum of its proper divisors (all divisors except itself). 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.
If the divisor sum is larger than the number it is abundant, like 12. If smaller it is deficient, like 8.
Two different numbers where each is the divisor sum of the other, such as 220 and 284.
Only 51 are known, and all are even. The first four are 6, 28, 496 and 8,128.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.