111 lines
2.9 KiB
Markdown
111 lines
2.9 KiB
Markdown
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---
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id: 5900f3c51000cf542c50fed6
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title: 'Problem 88: Product-sum numbers'
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challengeType: 5
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forumTopicId: 302203
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dashedName: problem-88-product-sum-numbers
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---
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# --description--
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A natural number, `N`, that can be written as the sum and product of a given set of at least two natural numbers, $\\{a_1, a_2, \ldots , a_k\\}$ is called a product-sum number: $N = a_1 + a_2 + \cdots + a_k = a_1 × a_2 × \cdots × a_k$.
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For example, 6 = 1 + 2 + 3 = 1 × 2 × 3.
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For a given set of size, `k`, we shall call the smallest N with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, `k` = 2, 3, 4, 5, and 6 are as follows.
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<div style='margin-left: 4em;'>
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<var>k</var>=2: 4 = 2 × 2 = 2 + 2<br>
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<var>k</var>=3: 6 = 1 × 2 × 3 = 1 + 2 + 3<br>
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<var>k</var>=4: 8 = 1 × 1 × 2 × 4 = 1 + 1 + 2 + 4<br>
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<var>k</var>=5: 8 = 1 × 1 × 2 × 2 × 2 = 1 + 1 + 2 + 2 + 2<br>
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<var>k</var>=6: 12 = 1 × 1 × 1 × 1 × 2 × 6 = 1 + 1 + 1 + 1 + 2 + 6
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</div><br>
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Hence for 2 ≤ `k` ≤ 6, the sum of all the minimal product-sum numbers is 4 + 6 + 8 + 12 = 30; note that `8` is only counted once in the sum.
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In fact, as the complete set of minimal product-sum numbers for 2 ≤ `k` ≤ 12 is $\\{4, 6, 8, 12, 15, 16\\}$, the sum is `61`.
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What is the sum of all the minimal product-sum numbers for 2 ≤ `k` ≤ `limit`?
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# --hints--
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`productSumNumbers(6)` should return a number.
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```js
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assert(typeof productSumNumbers(6) === 'number');
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```
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`productSumNumbers(6)` should return `30`.
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```js
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assert.strictEqual(productSumNumbers(6), 30);
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```
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`productSumNumbers(12)` should return `61`.
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```js
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assert.strictEqual(productSumNumbers(12), 61);
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```
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`productSumNumbers(300)` should return `12686`.
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```js
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assert.strictEqual(productSumNumbers(300), 12686);
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```
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`productSumNumbers(6000)` should return `2125990`.
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```js
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assert.strictEqual(productSumNumbers(6000), 2125990);
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```
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`productSumNumbers(12000)` should return `7587457`.
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```js
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assert.strictEqual(productSumNumbers(12000), 7587457);
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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 productSumNumbers(limit) {
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return true;
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}
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productSumNumbers(6);
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```
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# --solutions--
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```js
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function productSumNumbers(limit) {
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function getProductSums(curProduct, curSum, factorsCount, start) {
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const k = curProduct - curSum + factorsCount;
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if (k <= limit) {
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if (curProduct < minimalProductSums[k]) {
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minimalProductSums[k] = curProduct;
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}
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for (let i = start; i < Math.floor((limit / curProduct) * 2) + 1; i++) {
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getProductSums(curProduct * i, curSum + i, factorsCount + 1, i);
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}
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}
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}
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const minimalProductSums = new Array(limit + 1).fill(2 * limit);
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getProductSums(1, 1, 1, 2);
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const uniqueProductSums = [...new Set(minimalProductSums.slice(2))];
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let sum = 0;
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for (let i = 0; i < uniqueProductSums.length; i++) {
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sum += uniqueProductSums[i];
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}
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return sum;
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}
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```
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