Abundant Number Calculator

This abundant number calculator tells you whether a whole number is abundant, deficient or perfect, based on how its proper divisors add up. A number is abundant when the sum of its proper divisors, meaning every divisor except the number itself, is greater than the number; it is deficient when that sum falls short, and perfect on the rare occasion the divisors add up to exactly the number. Enter any whole number between 2 and 1,000,000,000 and the calculator instantly works out the sum of its proper divisors, states the abundance, which is the sum minus the number, and tells you which of the three categories it falls into. For example, entering 12 shows its proper divisors of 1, 2, 3, 4 and 6 sum to 16, giving an abundance of 4 and confirming 12 as the smallest abundant number. Results for very large numbers are shown with digit grouping so long strings of figures stay easy to read. The tool is handy for students studying number theory, programmers testing divisor algorithms, and anyone curious about the patterns in how numbers factor, since abundant numbers tend to cluster around multiples of small primes like 2, 3 and 5. Because it works through the exact integer algorithm rather than an approximation, every result is precise, no matter how large the number entered.

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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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