2.7 KiB
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;
}