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id	title	challengeType	forumTopicId	dashedName
5900f3b21000cf542c50fec5	Problem 70: Totient permutation	5	302183	problem-70-totient-permutation

--description--

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.

Interestingly, {\phi}(87109) = 79180, and it can be seen that 87109 is a permutation of 79180.

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.

--hints--

totientPermutation(10000) should return a number.

assert(typeof totientPermutation(10000) === 'number');

totientPermutation(10000) should return 4435.

assert.strictEqual(totientPermutation(10000), 4435);

totientPermutation(100000) should return 75841.

assert.strictEqual(totientPermutation(100000), 75841);

totientPermutation(500000) should return 474883.

assert.strictEqual(totientPermutation(500000), 474883);

totientPermutation(10000000) should return 8319823.

assert.strictEqual(totientPermutation(10000000), 8319823);

--seed--

--seed-contents--

function totientPermutation(limit) {

  return true;
}

totientPermutation(10000);

--solutions--

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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