HesapHanem

Perfect Number Checker

Check if a number is perfect.

Result

Perfect number
28

How it is calculated

The tool adds up all the proper divisors of n, meaning every divisor except n itself, and compares that sum s(n) with n. If s(n) = n the number is perfect; if s(n) > n it is abundant; if s(n) < n it is deficient. Divisors are found in pairs by testing every i up to the square root of n, so large values are quick. n must be a whole number from 1 to 100,000,000; decimals are cut to the integer part and values outside the range give no result. The number 1 has no proper divisors, so its sum is 0 and it is deficient.

Worked example

Take n = 28. Its divisors are 1, 2, 4, 7, 14 and 28. Leaving out 28 itself, the proper divisors add up to 1 + 2 + 4 + 7 + 14 = 28. The sum equals n, so the result is a sum of 28 and the status Perfect number. For comparison, 12 gives 1 + 2 + 3 + 4 + 6 = 16, which is larger, so 12 is abundant.

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Common mistakes

  • Including the number itself in the divisor sum; then every perfect number would look like double (28 gives 56).
  • Leaving out 1, which is a proper divisor of every number.
  • Thinking that abundant means large or deficient means small; 945 is odd and abundant, while a prime is always deficient.
  • Entering a decimal like 28.9 and expecting it to be treated as 29; it is truncated to 28.
  • Expecting large perfect numbers by brute force; they are very rare, and the tool stops at 100,000,000.

Sources

About this calculator

This calculator checks whether a number is perfect — equal to the sum of its own proper divisors, like 6 (1+2+3) or 28 (1+2+4+7+14) — and if not, whether it's abundant (divisors sum to more than the number) or deficient (divisors sum to less).

Perfect numbers have fascinated mathematicians since Euclid, and this tool is used by math students and hobbyists exploring number theory, since every known even perfect number is tied to a Mersenne prime.

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Comments (1)

  • Editorial team

    Within the range this tool accepts, up to 100,000,000, only five perfect numbers exist: 6, 28, 496, 8128 and 33,550,336. The next one is 8,589,869,056, which is beyond the limit. Anything that is not perfect is called abundant when its proper divisors sum to more than itself and deficient when they sum to less, and the number 1 counts as deficient. Are you hunting for patterns among abundant numbers?

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