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---
id: 5900f3b21000cf542c50fec5
title: 'Problem 70: Totient permutation'
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challengeType: 5
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forumTopicId: 302183
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dashedName: problem-70-totient-permutation
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---
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# --description--
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Euler's Totient function, ${\phi}(n)$ (sometimes called the phi function), is used to determine the number of positive numbers less than or equal to `n` which are relatively prime to `n` . For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, ${\phi}(9) = 6$. The number 1 is considered to be relatively prime to every positive number, so ${\phi}(1) = 1$.
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Interestingly, ${\phi}(87109) = 79180$, and it can be seen that 87109 is a permutation of 79180.
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Find the value of `n` , 1 < `n` < `limit` , for which ${\phi}(n)$ is a permutation of `n` and the ratio $\displaystyle\frac{n}{{\phi}(n)}$ produces a minimum.
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# --hints--
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`totientPermutation(10000)` should return a number.
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```js
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assert(typeof totientPermutation(10000) === 'number');
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```
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`totientPermutation(10000)` should return `4435` .
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```js
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assert.strictEqual(totientPermutation(10000), 4435);
```
`totientPermutation(100000)` should return `75841` .
```js
assert.strictEqual(totientPermutation(100000), 75841);
```
`totientPermutation(500000)` should return `474883` .
```js
assert.strictEqual(totientPermutation(500000), 474883);
```
`totientPermutation(10000000)` should return `8319823` .
```js
assert.strictEqual(totientPermutation(10000000), 8319823);
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```
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# --seed--
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## --seed-contents--
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```js
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function totientPermutation(limit) {
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return true;
}
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totientPermutation(10000);
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```
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# --solutions--
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```js
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function totientPermutation(limit) {
function getSievePrimes(max) {
const primes = [];
const primesMap = new Array(max).fill(true);
primesMap[0] = false;
primesMap[1] = false;
for (let i = 2; i < max ; i + = 2 ) {
if (primesMap[i]) {
primes.push(i);
for (let j = i * i; j < max ; j + = i ) {
primesMap[j] = false;
}
}
if (i === 2) {
i = 1;
}
}
return primes;
}
function sortDigits(number) {
return number.toString().split('').sort().join('');
}
function isPermutation(numberA, numberB) {
return sortDigits(numberA) === sortDigits(numberB);
}
const MAX_PRIME = 4000;
const primes = getSievePrimes(MAX_PRIME);
let nValue = 1;
let minRatio = Infinity;
for (let i = 1; i < primes.length ; i + + ) {
for (let j = i + 1; j < primes.length ; j + + ) {
const num = primes[i] * primes[j];
if (num > limit) {
break;
}
const phi = (primes[i] - 1) * (primes[j] - 1);
const ratio = num / phi;
if (minRatio > ratio & & isPermutation(num, phi)) {
nValue = num;
minRatio = ratio;
}
}
}
return nValue;
}
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```