Amicable Numbers Calculator: Finding Divisor-Sum Pairs Like 220 and 284
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
Point an amicable numbers calculator at 220 and it does two things in order. First it adds the proper divisors of 220 — 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. Then, and this is the half people skip, it turns around and adds the proper divisors of 284: 1 + 2 + 4 + 71 + 142 = 220. The sums swap. Each number is built out of the other one’s parts, and that mutual arithmetic is the entire definition of an amicable pair.
Only 23 such pairs start below 300,000, and the story of how they were found runs from a Baghdad manuscript in the ninth century to a sixteen-year-old with a notebook in 1866. What follows is that story, with the arithmetic that goes with it: the rule that manufactures pairs from three primes, the reason Euler could produce sixty of them while everyone before him managed three, and what happens when you keep taking divisor sums past the second step.
Why an Amicable Numbers Calculator Runs the Test Twice
Write s(n) for the aliquot sum — every divisor of n except n itself. Two distinct numbers a and b are amicable when s(a) = b and s(b) = a. Both conditions, not one. A single matching direction is the most common way a candidate falls over, and it happens constantly: s(95) = 25, but s(25) = 6, so 95 and 25 are nothing to each other.
There is a compact way to state the same test that also gives you a checksum. Since s(n) = σ(n) − n, where σ counts every divisor including n, amicability collapses to a single equation:
For 220 and 284 both divisor sums come out at 504, and 220 + 284 = 504. If you compute σ for each side and the two values disagree, stop — you do not have a pair, no matter how promising the first direction looked. Building σ from the prime factorisation instead of a divisor list is what keeps this fast on big inputs; the abundant number calculator walks through that bracket-multiplication trick in detail, and a prime factorization calculator gets you the input for it.
One more distinction worth keeping straight. If s(n) = n, the number is its own partner and it is called perfect, not amicable — 6, 28 and 496 all sit in that category. Amicable pairs need two different numbers. And if the sums land one step too far, as in s(48) = 76 and s(75) = 49, you have a betrothed pair instead: a near miss that is rarer than the real thing.
Thabit’s Rule: Three Primes and You Have a Pair
Around the year 850, Thabit ibn Qurra found a formula that manufactures amicable pairs out of nothing but a prime check. Pick an integer n > 1 and compute three candidates:
If all three are prime, then 2n·p·q and 2n·r are amicable. That is a remarkable amount of structure to get for free, and it is also where the trouble starts — all three have to be prime simultaneously, and they almost never are.
| n | p | q | r | Result |
|---|---|---|---|---|
| 2 | 5 | 11 | 71 | 220 & 284 |
| 3 | 11 | 23 | 287 = 7·41 | r composite — nothing |
| 4 | 23 | 47 | 1,151 | 17,296 & 18,416 |
| 5 | 47 | 95 = 5·19 | 4,607 | q composite — nothing |
| 6 | 95 = 5·19 | 191 | 18,431 | p composite — nothing |
| 7 | 191 | 383 | 73,727 | 9,363,584 & 9,437,056 |
Three hits in six attempts looks generous until you keep going: n = 2, 4 and 7 are the only values that have ever worked, and the search has run far past 20,000. The rule that gave the medieval world its second and third pairs has produced nothing since. Fermat rediscovered 17,296 and 18,416 in 1636 and Descartes turned up the seven-digit pair two years later — both, in effect, re-running Thabit’s formula without knowing it.
How Euler Went From Three Known Pairs to Sixty
By 1747 the world’s complete inventory of amicable pairs was three, and all three came out of the same formula. Then Euler took the rule apart. His generalisation drops the insistence on 2n and on consecutive exponents: look for pairs of the form a = m·p·q and b = m·r, where m is any common factor and p, q, r are primes to be determined. Feeding that into σ(a) = σ(b) = a + b turns the search into a system of equations in the unknown primes rather than a blind scan.
The 1750 catalogue that came out of it listed on the order of sixty new pairs at a stroke — counts in the literature differ slightly, and two of Euler’s entries were later shown to be wrong, which tells you something about how much hand computation was involved. Either way the known population went from 3 to more than 60 in a single publication. Euler had also broken the assumption that pairs must be even: 12,285 and 14,595, the smallest odd pair, belongs to this era of systematic searching rather than to the Thabit family, which only ever produces even numbers.
The modern count is past one billion pairs, all of them found by machine. The largest have tens of thousands of digits. What has not happened in 2,000 years is a proof that the supply is infinite — nobody knows whether amicable pairs run out.
The Pair That Sat in Plain Sight for 800 Years
Here is the part that ought to be embarrassing for the whole enterprise. The second-smallest amicable pair is 1,184 and 1,210. Four digits. Well inside the reach of anyone doing arithmetic by hand, and squarely inside the range Euler swept with his equations. It was found in 1866 by B. Nicolò I. Paganini, an Italian schoolboy of sixteen, who appears to have simply checked small numbers one at a time.
Verify it in two lines. 1,184 = 25 × 37, so σ = 63 × 38 = 2,394 and s(1,184) = 2,394 − 1,184 = 1,210. Going back: 1,210 = 2 × 5 × 112, so σ = 3 × 6 × 133 = 2,394 and s(1,210) = 2,394 − 1,210 = 1,184. Both σ values equal 2,394, which is exactly 1,184 + 1,210 — the checksum from earlier, holding.
The lesson generalises past this one pair. Euler’s method was a generator: it produces numbers of a particular algebraic shape, and anything outside that shape is invisible to it, no matter how small. Exhaustive search and clever formulas find different things, which is why the range mode in the calculator above is worth running even over intervals that famous mathematicians have already crossed.
What the First Ten Pairs Have in Common
Line up the small pairs and some regularities jump out immediately, along with one that has resisted proof for centuries.
| Pair | Ratio b/a | gcd | Parity |
|---|---|---|---|
| 220 & 284 | 1.291 | 4 | even |
| 1,184 & 1,210 | 1.022 | 2 | even |
| 2,620 & 2,924 | 1.116 | 4 | even |
| 5,020 & 5,564 | 1.108 | 4 | even |
| 6,232 & 6,368 | 1.022 | 8 | even |
| 10,744 & 10,856 | 1.010 | 8 | even |
| 12,285 & 14,595 | 1.188 | 1,365 | odd |
| 17,296 & 18,416 | 1.065 | 16 | even |
| 63,020 & 76,084 | 1.207 | 4 | even |
| 66,928 & 66,992 | 1.001 | 16 | even |
Every one of them shares a factor. That is not a theorem — it is an observation that has survived a billion examples. If a coprime amicable pair exists, it has been shown that its two members must multiply to more than 1067, which is why a greatest common factor calculator run on any pair you find will keep returning something bigger than 1.
Parity is the same kind of stubborn pattern: both members are always even or always odd, never one of each, and again nobody has proved it must be so. The ratios cluster tight — 66,928 and 66,992 differ by 64, about a tenth of one percent — because both numbers have to carry roughly the same divisor weight to name each other. And the odd pairs, when they turn up, tend to be multiples of 3 and 5 with an Euler phi structure quite unlike their even cousins; 12,285 = 33 × 5 × 7 × 13 is typical of the type.
Past the Pair: Chains, Sociable Numbers and 276
Nothing forces you to stop after two steps. Keep applying s and you get an aliquot sequence: n, s(n), s(s(n)), and onward. Seen this way, perfect numbers and amicable pairs stop looking like separate curiosities and become the two shortest cases of one phenomenon — a cycle.
| Cycle length | Name | Smallest example |
|---|---|---|
| 1 | Perfect number | 6 → 6 |
| 2 | Amicable pair | 220 → 284 → 220 |
| 5 | Sociable chain | 12,496 → 14,288 → … |
| 28 | Sociable chain | 14,316 → 19,116 → … |
Paul Poulet turned up both sociable chains in 1918, and the 28-link one is still the longest known. Notice what is missing: no cycle of length 3 has ever been found, and none of length 4 below any searched bound either. Why the gap between 2 and 5 exists is unexplained.
Chains can also arrive from outside. Start at 562 — a number with nothing special about it — and s(562) = 284, which drops you straight into the 220/284 loop. Most starting points do the opposite and drain away: 95 → 25 → 6 and then sits on the perfect number forever, while anything prime collapses in one step to 1 and then to 0.
And then there is 276. Its sequence has been pushed past two thousand terms and two hundred digits without either closing into a cycle or coming back down. Four other numbers under 1,000 — 552, 564, 660 and 966 — behave the same way; collectively they are known as the Lehmer five. Whether every aliquot sequence eventually terminates or cycles is the Catalan–Dickson conjecture, still open. The chain view in Find the partner above stops when a term passes ten digits, which is exactly where those five leave the reach of a browser and the question becomes somebody else’s computing cluster. The full catalogue of known pairs lives in OEIS sequence A259180, and Sergei Chernykh’s Amicable Pairs List holds the billion-plus that machines have found since.
