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Amicable Numbers Calculator

Work out amicable numbers instantly with clear inputs, formula shown and shareable results.

About Amicable Numbers Calculator

Two numbers are amicable when each equals the sum of the other's proper divisors. The smallest pair is 220 and 284, known since antiquity. A number equal to its own aliquot sum is perfect rather than amicable.

Formula

Amicable pair

s(a) = b and s(b) = a with a ≠ b, where s is the sum of proper divisors

Worked example

With the default values below, Amicable Numbers Calculator returns Result of 220 is amicable with 284. Change any input above and every figure here is recalculated in your browser.

Inputs used

Number to testminimum 1
220
Search for amicable pairs up toaccepts 2–20,000
1,500

Results

Result
220 is amicable with 284
Sum of proper divisors
284
Sum of the partner's proper divisors
220
Amicable pairs in range
220 & 284, 1184 & 1210

Step by step

  1. 1Values used
  2. 2Amicable pair
  3. 3Result= 220 is amicable with 284
  4. 4Sum of proper divisors= 284
  5. 5Sum of the partner's proper divisors= 220
  6. 6Amicable pairs in range= 220 & 284, 1184 & 1210

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

Is 220 amicable?

Yes, with 284. The proper divisors of 220 sum to 284 and those of 284 sum to 220.

What is the next pair?

1184 and 1210, found by Paganini in 1866 — remarkably overlooked by earlier mathematicians.

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