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---
title: Amicable pairs
id: 5949b579404977fbaefcd737
challengeType: 5
---
## Description
< section id = 'description' >
Two integers $N$ and $M$ are said to be < a href = "https://en.wikipedia.org/wiki/Amicable numbers" title = "wp: Amicable numbers" > amicable pairs</ a > if $N \neq M$ and the sum of the < a href = "http://rosettacode.org/wiki/Proper divisors" title = "Proper divisors" > proper divisors</ a > of $N$ ($\mathrm{sum}(\mathrm{propDivs}(N))$) $= M$ as well as $\mathrm{sum}(\mathrm{propDivs}(M)) = N$.
Example:
1184 and 1210 are an amicable pair, with proper divisors:
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1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 and
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1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 respectively.
Task:
Calculate and show here the Amicable pairs below 20,000 (there are eight).
Related tasks
< a href = "http://rosettacode.org/wiki/Proper divisors" title = "Proper divisors" > Proper divisors< / a >
< a href = "http://rosettacode.org/wiki/Abundant, deficient and perfect number classifications" title = "Abundant, deficient and perfect number classifications" > Abundant, deficient and perfect number classifications< / a >
< a href = "http://rosettacode.org/wiki/Aliquot sequence classifications" title = "Aliquot sequence classifications" > Aliquot sequence classifications< / a > and its amicable classification.
< / section >
## Instructions
< section id = 'instructions' >
< / section >
## Tests
< section id = 'tests' >
```yml
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tests:
- text: < code > amicablePairsUpTo</ code > is a function.
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testString: assert(typeof amicablePairsUpTo === 'function', '< code > amicablePairsUpTo< / code > is a function.');
- text: < code > amicablePairsUpTo(300)</ code > should return < code > [[220,284]]</ code > .
testString: assert.deepEqual(amicablePairsUpTo(300), answer300, '< code > amicablePairsUpTo(300)< / code > should return < code > [[220,284]]< / code > .');
- text: < code > amicablePairsUpTo(3000)</ code > should return < code > [[220,284],[1184,1210],[2620,2924]]</ code > .
testString: assert.deepEqual(amicablePairsUpTo(3000), answer3000, '< code > amicablePairsUpTo(3000)< / code > should return < code > [[220,284],[1184,1210],[2620,2924]]< / code > .');
- text: < code > amicablePairsUpTo(20000)</ code > should return < code > [[220,284],[1184,1210],[2620,2924],[5020,5564],[6232,6368],[10744,10856],[12285,14595],[17296,18416]]</ code > .
testString: assert.deepEqual(amicablePairsUpTo(20000), answer20000, '< code > amicablePairsUpTo(20000)< / code > should return < code > [[220,284],[1184,1210],[2620,2924],[5020,5564],[6232,6368],[10744,10856],[12285,14595],[17296,18416]]< / code > .');
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```
< / section >
## Challenge Seed
< section id = 'challengeSeed' >
< div id = 'js-seed' >
```js
function amicablePairsUpTo (maxNum) {
// Good luck!
return true;
}
```
< / div >
### After Test
< div id = 'js-teardown' >
```js
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const answer300 = [[220, 284]];
const answer3000 = [
[220, 284],
[1184, 1210],
[2620, 2924]
];
const answer20000 = [
[220, 284],
[1184, 1210],
[2620, 2924],
[5020, 5564],
[6232, 6368],
[10744, 10856],
[12285, 14595],
[17296, 18416]
];
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```
< / div >
< / section >
## Solution
< section id = 'solution' >
```js
// amicablePairsUpTo :: Int -> [(Int, Int)]
function amicablePairsUpTo (maxNum) {
return range(1, maxNum)
.map(x => properDivisors(x)
.reduce((a, b) => a + b, 0))
.reduce((a, m, i, lst) => {
const n = i + 1;
return (m > n) & & lst[m - 1] === n ?
a.concat([
[n, m]
]) : a;
}, []);
}
// properDivisors :: Int -> [Int]
function properDivisors (n) {
if (n < 2 ) return [ ] ;
const rRoot = Math.sqrt(n);
const intRoot = Math.floor(rRoot);
const blnPerfectSquare = rRoot === intRoot;
const lows = range(1, intRoot)
.filter(x => (n % x) === 0);
return lows.concat(lows.slice(1)
.map(x => n / x)
.reverse()
.slice(blnPerfectSquare | 0));
}
// Int -> Int -> Maybe Int -> [Int]
function range (m, n, step) {
const d = (step || 1) * (n >= m ? 1 : -1);
return Array.from({
length: Math.floor((n - m) / d) + 1
}, (_, i) => m + (i * d));
}
```
< / section >