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Sum Of Divisors Calculator

Work out sum of divisors instantly with clear inputs, formula shown and shareable results.

About Sum Of Divisors Calculator

σ(n) sums every divisor including n itself. Comparing the aliquot sum σ(n) − n with n classifies the number: equal is perfect, greater is abundant and smaller is deficient.

Formula

Sigma from factorisation

σ(p₁^e₁ ⋯ pₖ^eₖ) = Π (p^(e+1) − 1)/(p − 1)

Worked example

With the default values below, Sum Of Divisors Calculator returns Sum of all divisors σ(n) of 56. Change any input above and every figure here is recalculated in your browser.

Inputs used

Numberminimum 1
28
Compare withminimum 1
12

Results

Sum of all divisors σ(n)
56
Sum of proper divisors
28
Classification
Perfect number
σ of the comparison number
28

Step by step

  1. 1Values used
  2. 2Sigma from factorisation
  3. 3Sum of all divisors σ(n)= 56
  4. 4Sum of proper divisors= 28
  5. 5Classification= Perfect number
  6. 6σ of the comparison number= 28

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Why is 28 perfect?

Its proper divisors 1, 2, 4, 7 and 14 sum to exactly 28.

Are there odd perfect numbers?

None is known, and if one exists it must exceed 10¹⁵⁰⁰. It is one of the oldest open problems in mathematics.

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