Abundant Number Calculator

This abundant number calculator checks whether a whole number is abundant, meaning its proper divisors add up to more than the number itself. Enter a value and the calculator gives the sum of the proper divisors, the abundance (how much the sum exceeds the number), and classifies it as abundant, perfect or deficient. Abundant numbers are common among the even numbers; 12 is the smallest, with divisors 1, 2, 3, 4 and 6 summing to 16. They contrast with deficient numbers, where the divisors fall short, and the rare perfect numbers, where they hit the total exactly. 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

Abundance (sum minus n)

The formula

Add the proper divisors of n (all divisors except n). If the sum is greater than n, the number is abundant, and the abundance is the sum minus n. Equal means perfect, less means deficient.

Worked example

The proper divisors of 12 are 1, 2, 3, 4 and 6, adding to 16, which is more than 12, so 12 is abundant with abundance 4. Enter 12 to confirm.

Frequently asked questions

What is an abundant number?

A number whose proper divisors add up to more than the number itself, such as 12, 18 and 20.

What is the smallest abundant number?

12, with proper divisors summing to 16.

Are most numbers abundant or deficient?

Deficient numbers are more common overall, but abundant numbers are plentiful, especially among multiples of small primes.

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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