Perfect Number Calculator – Perfect, Abundant, or Deficient?

Perfect number calculator that checks if a positive integer is perfect, abundant, or deficient. Lists proper divisors, divisor sum, and shows known perfect numbers and their properties.

Full original guide (expanded)

Perfect Number Calculator – Perfect, Abundant, or Deficient?

Enter a positive integer to check if it is a perfect number and see its proper divisors, divisor sum, and classification as perfect, abundant, or deficient. Optionally search for all perfect numbers in a range.

Built for number theory students, teachers, and curious learners who want transparent, step-by-step divisor analysis.

Perfect number checker

Divisor-based classification

Choose a mode, enter your integer(s), and click Calculate. The tool computes proper divisors, their sum, and a clear classification.

Mode

Enter a positive integer. For very large n the computation may take longer; this tool is optimized for classroom-sized examples.

Divisor search up to ⌊√n⌋ with symmetry (d, n/d) for efficiency.

Results, divisor lists, and classifications will appear here after you click Calculate.

What is a perfect number?

A positive integer \(n\) is called a perfect number if it equals the sum of its proper divisors, that is, all positive divisors less than \(n\) itself. Formally, if \(\sigma(n)\) is the sum of all positive divisors of \(n\), then \(n\) is perfect when

\[ \sigma(n) - n = n \quad\Longleftrightarrow\quad \sum_{\substack{d \mid n\\ d < n}} d = n. \]

The first perfect numbers are \(6, 28, 496, 8128,\dots\). For example, the proper divisors of 28 are \(1, 2, 4, 7, 14\), and \(1 + 2 + 4 + 7 + 14 = 28\).

Perfect, abundant, and deficient numbers

The same divisor-sum test also gives two related classifications:

  • Perfect: \(\sum_{d \mid n,\, d < n} d = n\).
  • Abundant: \(\sum_{d \mid n,\, d < n} d > n\) (the number has “too much” divisor mass).
  • Deficient: \(\sum_{d \mid n,\, d < n} d < n\) (the divisor sum falls short of the number).

Many everyday integers such as 8, 10, and 11 are deficient; abundant numbers like 12 and 18 appear regularly; perfect numbers are extremely rare.

Even perfect numbers and Mersenne primes

All known perfect numbers are even and have a special form discovered by Euclid and Euler:

\[ n = 2^{p-1}\,\bigl(2^p - 1\bigr), \]

where \(2^p - 1\) is a Mersenne prime, a prime number of the form \(2^p - 1\).

Every time a new Mersenne prime \(2^p - 1\) is found, it generates a new even perfect number. Modern record perfect numbers are discovered using large-scale distributed computing projects.

Do odd perfect numbers exist?

No one knows. Despite centuries of work, no odd perfect number has ever been found. At the same time, there is no proof that odd perfect numbers cannot exist; it remains a deep open problem in number theory.

How the calculator finds proper divisors

For a given \(n\), the calculator performs an efficient divisor search:

  • It always includes 1 as a proper divisor (for \(n > 1\)).
  • It loops over integers \(d\) from 2 up to \(\lfloor\sqrt{n}\rfloor\).
  • Whenever \(d\) divides \(n\), it adds both \(d\) and \(n/d\) (when distinct) to the divisor list.
  • It removes \(n\) itself from the list, sorts the remaining divisors, and sums them.

This algorithm runs in \(O(\sqrt{n})\) time, which is efficient for the typical ranges used in teaching and exploratory work.

Perfect number calculator – FAQ

Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[\sigma(n) - n = n \quad\Longleftrightarrow\quad \sum_{\substack{d \mid n\\ d < n}} d = n.\]
\sigma(n) - n = n \quad\Longleftrightarrow\quad \sum_{\substack{d \mid n\\ d < n}} d = n.
Formula (extracted LaTeX)
\[n = 2^{p-1}\,\bigl(2^p - 1\bigr),\]
n = 2^{p-1}\,\bigl(2^p - 1\bigr),
Formula (extracted LaTeX)
\[','\\]
','\
Formula (extracted text)
\[ \sigma(n) - n = n \quad\Longleftrightarrow\quad \sum_{\substack{d \mid n\\ d < n}} d = n. \]
Formula (extracted text)
\[ n = 2^{p-1}\,\bigl(2^p - 1\bigr), \] where \(2^p - 1\) is a Mersenne prime, a prime number of the form \(2^p - 1\).
Variables and units
  • No variables provided in audit spec.
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
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Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft
Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog
  • 0.1.0-draft — 2026-01-19: Initial draft (review required).