2f901f9ffe
* fix: rework challenge to use argument in function * fix: add solution * fix: position equations evenly between paragraphs
132 lines
2.9 KiB
Markdown
132 lines
2.9 KiB
Markdown
---
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id: 5900f3c51000cf542c50fed8
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title: 'Problem 87: Prime power triples'
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challengeType: 5
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forumTopicId: 302201
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dashedName: problem-87-prime-power-triples
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---
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# --description--
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The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is `28`. In fact, there are exactly four numbers below fifty that can be expressed in such a way:
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<div style='margin-left: 4em;'>
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28 = 2<sup>2</sup> + 2<sup>3</sup> + 2<sup>4</sup><br>
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33 = 3<sup>2</sup> + 2<sup>3</sup> + 2<sup>4</sup><br>
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49 = 5<sup>2</sup> + 2<sup>3</sup> + 2<sup>4</sup><br>
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47 = 2<sup>2</sup> + 3<sup>3</sup> + 2<sup>4</sup>
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</div><br>
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How many numbers below `n` can be expressed as the sum of a prime square, prime cube, and prime fourth power?
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# --hints--
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`primePowerTriples(50)` should return a number.
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```js
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assert(typeof primePowerTriples(50) === 'number');
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```
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`primePowerTriples(50)` should return `4`.
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```js
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assert.strictEqual(primePowerTriples(50), 4);
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```
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`primePowerTriples(10035)` should return `684`.
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```js
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assert.strictEqual(primePowerTriples(10035), 684);
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```
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`primePowerTriples(500000)` should return `18899`.
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```js
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assert.strictEqual(primePowerTriples(500000), 18899);
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```
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`primePowerTriples(5000000)` should return `138932`.
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```js
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assert.strictEqual(primePowerTriples(5000000), 138932);
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```
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`primePowerTriples(50000000)` should return `1097343`.
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```js
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assert.strictEqual(primePowerTriples(50000000), 1097343);
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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 primePowerTriples(n) {
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return true;
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}
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primePowerTriples(50);
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```
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# --solutions--
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```js
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function primePowerTriples(n) {
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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 = 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 getPowersSummed(numbers, powers, limit, curSum) {
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if (curSum >= limit) {
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return [];
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} else if (powers.length === 0) {
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return [curSum];
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}
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const powersSummed = [];
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const curPower = powers[0];
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const powersLeft = powers.slice(1);
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for (let i = 0; i < numbers.length; i++) {
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const curNumber = numbers[i];
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const nextSum = curSum + curNumber ** curPower;
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if (nextSum >= limit) {
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return powersSummed;
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}
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const result = getPowersSummed(
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numbers,
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powersLeft,
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limit,
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curSum + curNumber ** curPower
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);
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powersSummed.push(...result);
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}
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return powersSummed;
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}
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const maximumBaseNumber = Math.floor(Math.sqrt(n - 2 ** 3 - 2 ** 4)) + 1;
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const primes = getSievePrimes(maximumBaseNumber);
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const uniqueSums = new Set(getPowersSummed(primes, [2, 3, 4], n, 0));
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return uniqueSums.size;
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}
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```
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