119 lines
2.7 KiB
Markdown
119 lines
2.7 KiB
Markdown
---
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id: 5900f3b21000cf542c50fec5
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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);
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```
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`totientPermutation(100000)` should return `75841`.
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```js
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assert.strictEqual(totientPermutation(100000), 75841);
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```
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`totientPermutation(500000)` should return `474883`.
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```js
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assert.strictEqual(totientPermutation(500000), 474883);
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```
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`totientPermutation(10000000)` should return `8319823`.
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```js
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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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}
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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) {
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function getSievePrimes(max) {
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const primes = [];
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const primesMap = new Array(max).fill(true);
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primesMap[0] = false;
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primesMap[1] = false;
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for (let i = 2; i < max; i += 2) {
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if (primesMap[i]) {
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primes.push(i);
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for (let j = i * i; j < max; j += i) {
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primesMap[j] = false;
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}
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}
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if (i === 2) {
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i = 1;
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}
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}
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return primes;
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}
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function sortDigits(number) {
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return number.toString().split('').sort().join('');
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}
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function isPermutation(numberA, numberB) {
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return sortDigits(numberA) === sortDigits(numberB);
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}
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const MAX_PRIME = 4000;
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const primes = getSievePrimes(MAX_PRIME);
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let nValue = 1;
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let minRatio = Infinity;
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for (let i = 1; i < primes.length; i++) {
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for (let j = i + 1; j < primes.length; j++) {
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const num = primes[i] * primes[j];
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if (num > limit) {
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break;
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}
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const phi = (primes[i] - 1) * (primes[j] - 1);
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const ratio = num / phi;
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if (minRatio > ratio && isPermutation(num, phi)) {
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nValue = num;
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minRatio = ratio;
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}
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}
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}
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return nValue;
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}
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```
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