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---
id: 5900f3c51000cf542c50fed6
title: 'Problem 88: Product-sum numbers'
challengeType: 5
forumTopicId: 302203
dashedName: problem-88-product-sum-numbers
---
# --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.
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.
< div style = 'margin-left: 4em;' >
< var > k< / var > =2: 4 = 2 × 2 = 2 + 2< br >
< var > k< / var > =3: 6 = 1 × 2 × 3 = 1 + 2 + 3< br >
< var > k< / var > =4: 8 = 1 × 1 × 2 × 4 = 1 + 1 + 2 + 4< br >
< var > k< / var > =5: 8 = 1 × 1 × 2 × 2 × 2 = 1 + 1 + 2 + 2 + 2< br >
< 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);
```
`productSumNumbers(12)` should return `61` .
```js
assert.strictEqual(productSumNumbers(12), 61);
```
`productSumNumbers(300)` should return `12686` .
```js
assert.strictEqual(productSumNumbers(300), 12686);
```
`productSumNumbers(6000)` should return `2125990` .
```js
assert.strictEqual(productSumNumbers(6000), 2125990);
```
`productSumNumbers(12000)` should return `7587457` .
```js
assert.strictEqual(productSumNumbers(12000), 7587457);
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```
# --seed--
## --seed-contents--
```js
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function productSumNumbers(limit) {
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return true;
}
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productSumNumbers(6);
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```
# --solutions--
```js
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function productSumNumbers(limit) {
function getProductSums(curProduct, curSum, factorsCount, start) {
const k = curProduct - curSum + factorsCount;
if (k < = limit) {
if (curProduct < minimalProductSums [ k ] ) {
minimalProductSums[k] = curProduct;
}
for (let i = start; i < Math.floor ( ( limit / curProduct ) * 2 ) + 1 ; i + + ) {
getProductSums(curProduct * i, curSum + i, factorsCount + 1, i);
}
}
}
const minimalProductSums = new Array(limit + 1).fill(2 * limit);
getProductSums(1, 1, 1, 2);
const uniqueProductSums = [...new Set(minimalProductSums.slice(2))];
let sum = 0;
for (let i = 0; i < uniqueProductSums.length ; i + + ) {
sum += uniqueProductSums[i];
}
return sum;
}
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```