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Abundant Number Checker

Enter a whole number to see its proper divisor sum and whether it's abundant.

How it is calculated

The tool adds up every divisor of n except n itself (the proper divisors, which include 1), giving s(n). It then reports three numbers: the sum s(n), a flag that is 1 when s(n) > n and 0 otherwise, and the abundance s(n) − n. The sign of the abundance tells you the class: positive means abundant, zero means perfect, negative means deficient. The input must be a whole number from 1 to 100 000 000; decimals, zero, negatives and larger values return no result. For n = 1 the sum is 0, so 1 is deficient. The divisors themselves are not listed, only their total.

Worked example

Take n = 20. Step 1: its proper divisors are 1, 2, 4, 5 and 10. Step 2: add them, 1 + 2 + 4 + 5 + 10 = 22. Step 3: compare with 20. Since 22 is larger, the flag is 1 (abundant) and the abundance is 22 − 20 = 2. The tool prints a sum of proper divisors of 22, abundant flag 1, abundance 2.

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Common mistakes

  • Including the number itself in the divisor sum. That sum is the sigma function, which is always larger by n; the tool excludes n.
  • Forgetting 1 as a divisor. It always counts, and for a prime it is the only proper divisor.
  • Reading the flag 0 as "perfect". A flag of 0 covers both perfect and deficient numbers; check the abundance: exactly 0 is perfect (such as 6), negative is deficient.
  • Entering a decimal such as 12.5 or a number above 100 million and expecting an answer; only whole numbers in range are accepted.

About this calculator

A number is called abundant (or excessive) when the sum of its proper divisors — every divisor except the number itself — exceeds the number: 12 is the smallest example, since its proper divisors 1, 2, 3, 4, and 6 add up to 16. This classification sits alongside perfect numbers, whose proper divisors sum to exactly the number (like 6 = 1 + 2 + 3), and deficient numbers, whose proper divisors sum to less than the number — a three-way split of the integers first studied systematically by the ancient Greek mathematician Nicomachus of Gerasa around 100 CE.

The concept lives mainly in number theory and recreational mathematics today: it underlies results like the fact that every multiple of an abundant number is itself abundant, and it appears in classroom introductions to divisor functions and the sigma function used to compute the sum of divisors.

This calculator takes a whole number, adds up its proper divisors, and returns that sum, an abundant flag (1 if the number is abundant, 0 if not), and the abundance — the divisor sum minus the number. A positive abundance means the number is abundant, zero means it is perfect, and a negative value means it is deficient.

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