Amicable Number Finder

s(a)=b, s(b)=a

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About Amicable Number Finder

An amicable number finder that checks if two numbers form an amicable pair: s(a)=b and s(b)=a. Searches for amicable pairs in a range. Shows divisor analysis for both numbers. All calculations are client-side.

Amicable Number Finder Features

  • Pair check
  • Range search
  • Divisor analysis
  • Known pairs
  • Aliquot chain
Amicable numbers: s(a)=b and s(b)=a where s(n) = sum of proper divisors. Smallest pair: (220, 284). s(220) = 1+2+4+5+10+11+20+22+44+55+110 = 284, s(284) = 1+2+4+71+142 = 220. Known since Pythagoras.

How to Use

Enter a number or range:

  • Check: Is (a,b) amicable?
  • Search: Find pairs in range

History

Pythagoras knew (220,284). Thābit ibn Qurra (9th century) found a formula. Euler discovered ~60 pairs. Paganini (age 16!) found (1184,1210) that Euler missed.

Related Types

  • Perfect: s(n)=n (amicable with itself)
  • Sociable: cycle of length k>2
  • Aspiring: aliquot chain reaches a cycle

Step-by-Step Instructions

  1. 1Enter n.
  2. 2Check if amicable.
  3. 3Find its pair.
  4. 4Search range.
  5. 5View divisors.

Amicable Number Finder — Frequently Asked Questions

How many amicable pairs are known?+

Over 12 million pairs are known as of 2024. The smallest is (220, 284). Most known pairs were found by computer search. It's unknown whether infinitely many exist, though this is widely believed.

Is there a formula for amicable numbers?+

Thābit's rule (9th century): if p=3·2ⁿ⁻¹−1, q=3·2ⁿ−1, r=9·2²ⁿ⁻¹−1 are all prime, then (2ⁿpq, 2ⁿr) is amicable. This works for n=2,4,7 giving (220,284), (17296,18416), and others. But most pairs aren't from this formula.

Are all amicable numbers even?+

No! The first odd amicable pair is (12285, 14595), found by way of exhaustive search. Most known pairs are even-even. No even-odd pair has been found, and it's conjectured none exist.

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