Perfect, Abundant and Deficient Number Checker

Classify a positive integer by comparing it with the exact sum of its proper positive divisors. See the divisor list, prime factorization, and calculation steps. Your input stays in your browser.

Enter a positive integer

Commas, spaces, or underscores may group digits in threes. Zero, negative numbers, and decimals are outside this checker’s positive-integer definition.

Classification result

Your result will appear here.

Try 28: its proper divisors are 1, 2, 4, 7, and 14, which sum to 28, so it is perfect.

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Perfect, abundant, and deficient numbers

Let s(n) be the sum of the positive divisors of n other than n itself. These are its proper positive divisors. Comparing s(n) with n gives exactly one classification:

Perfect: s(n) = n

Abundant: s(n) > n

Deficient: s(n) < n

The proper-divisor sum is also called the aliquot sum. Because the full divisor sum σ(n) includes n, the calculator uses s(n) = σ(n) − n.

Examples of each classification

Perfect: 28

1 + 2 + 4 + 7 + 14 = 28

The proper-divisor sum equals the number.

Abundant: 12

1 + 2 + 3 + 4 + 6 = 16

The sum exceeds 12 by 4.

Deficient: 16

1 + 2 + 4 + 8 = 15

The sum falls short of 16 by 1.

How the checker calculates the result

  1. Factor n into prime powers: n = p₁a₁p₂a₂⋯pₖaₖ.
  2. Calculate the full positive-divisor sum with σ(n) = ∏(1 + pᵢ + pᵢ² + ⋯ + pᵢaᵢ).
  3. Subtract n to get the proper-divisor sum: s(n) = σ(n) − n.
  4. Compare s(n) with n to assign the classification.

Exactness and limits: calculations use exact BigInt arithmetic. Inputs are limited to 1 trillion, and factorization runs in a background worker with a five-second limit so unusually difficult inputs do not leave the page unresponsive.

Number classification FAQ

What is a perfect number?

A perfect number equals the sum of its proper positive divisors. For example, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.

What is an abundant number?

An abundant number has a proper-divisor sum greater than itself. For example, the proper divisors of 12 sum to 16.

What is a deficient number?

A deficient number has a proper-divisor sum less than itself. Every prime is deficient because its only proper positive divisor is 1.

Is 1 perfect, abundant, or deficient?

One is deficient. It has no proper positive divisors, so its proper-divisor sum is 0.

Are all perfect numbers even?

Every perfect number currently known is even. Whether an odd perfect number exists remains an open problem.

What is the abundance or deficiency?

The signed difference s(n) − n measures how far the proper-divisor sum is above or below the number. A positive difference is abundance, a negative difference is deficiency, and zero indicates a perfect number.

Is my input private?

Yes. Factorization and classification happen entirely in your browser. The checker does not upload, store, or add the number to the page URL.

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