HesapHanem

Abundant / Perfect / Deficient Number Checker

Classify a number by its aliquot sum.

How it is calculated

The tool adds up all proper divisors of n, meaning every positive divisor smaller than n itself (the aliquot sum s(n)), and compares it with n. If s(n) > n the number is Abundant, if s(n) = n it is Perfect, and if s(n) < n it is Deficient. It finds divisors in pairs by testing every i up to √n, so even large numbers are quick.

The input is treated as a whole number between 1 and 100,000,000; anything outside this range, such as 0, a negative number or 100,000,001, gives no result. Decimals are cut to the integer part. The number 1 has no proper divisors, so its aliquot sum is 0 and it is reported as Deficient.

Worked example

Take n = 60. Its proper divisors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. Their sum is 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 = 108. Since 108 is greater than 60, the number is Abundant, with an aliquot sum of 108.

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Common mistakes

  • Including the number itself in the divisor sum; for 6 that would give 12 and hide that 6 is perfect (1 + 2 + 3 = 6).
  • Mixing this up with the sigma function σ(n): σ(n) counts n as well, so a number is perfect when σ(n) = 2n, while this tool compares the sum without n to n.
  • Assuming even numbers are always abundant or odd ones always deficient; 12 is abundant but 945 is the smallest odd abundant number, and 8 is an even deficient number.
  • Entering a decimal such as 28.9, which is cut to 28 rather than rounded.
  • Entering numbers above 100,000,000 and wondering why nothing appears; the tool has a hard upper limit.

About this calculator

This tool classifies a whole number by comparing it to the sum of its proper divisors (its "aliquot sum" — every divisor except the number itself). A number is perfect when the aliquot sum equals the number (6 = 1+2+3, 28 = 1+2+4+7+14), abundant when the sum exceeds it (12: 1+2+3+4+6 = 16), and deficient when the sum falls short (8: 1+2+4 = 7). Most numbers are deficient; abundant numbers become more common as numbers grow, and perfect numbers are rare — only 52 are known, and every even one corresponds to a Mersenne prime via the Euclid-Euler theorem.

This classification dates back to ancient Greek number theory (Euclid, Nicomachus) and remains a staple of recreational mathematics and introductory number theory courses. Math teachers use it to illustrate divisor functions, programmers use it to practice algorithm efficiency (finding divisors up to √n), and puzzle enthusiasts use it to explore related curiosities like amicable and sociable numbers.

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Comments (1)

  • Editorial team

    The tool shows the aliquot sum, so you can read how far a number sits from perfection yourself: 12 has 16 (4 above), while 945, the smallest odd abundant number, has 975 (30 above). The input range runs to 100,000,000, which includes the fifth perfect number, 33,550,336. Note that 1 counts as deficient, because its aliquot sum is 0. Would a separate line for the difference help you?

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