63 lines
1.8 KiB
Markdown
63 lines
1.8 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`<sub>1</sub>, `a`<sub>2</sub>, ... , `a`<sub>k</sub>} is called a product-sum number: N = `a`<sub>1</sub> + `a`<sub>2</sub> + ... + `a`<sub>k</sub> = `a`<sub>1</sub> × `a`<sub>2</sub> × ... × `a`<sub>k</sub>.
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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>
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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`≤12000?
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# --hints--
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`productSumNumbers()` should return a number.
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```js
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assert(typeof productSumNumbers() === 'number');
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```
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`productSumNumbers()` should return 7587457.
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```js
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assert.strictEqual(productSumNumbers(), 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() {
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return true;
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}
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productSumNumbers();
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```
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# --solutions--
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```js
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// solution required
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```
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