---
id: 5900f3c51000cf542c50fed6
challengeType: 5
title: 'Problem 88: Product-sum numbers'
forumTopicId: 302203
---

## Description
<section id='description'>

A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, {<var>a</var><sub>1</sub>, <var>a</var><sub>2</sub>, ... , <var>a</var><sub>k</sub>} is called a product-sum number: N = <var>a</var><sub>1</sub> + <var>a</var><sub>2</sub> + ... + <var>a</var><sub>k</sub> = <var>a</var><sub>1</sub> × <var>a</var><sub>2</sub> × ... × <var>a</var><sub>k</sub>.

For example, 6 = 1 + 2 + 3 = 1 × 2 × 3.

For a given set of size, <var>k</var>, we shall call the smallest N with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, <var>k</var> = 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
</div>

Hence for 2≤<var>k</var>≤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.

In fact, as the complete set of minimal product-sum numbers for 2≤<var>k</var>≤12 is {4, 6, 8, 12, 15, 16}, the sum is 61.

What is the sum of all the minimal product-sum numbers for 2≤<var>k</var>≤12000?

</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>productSumNumbers()</code> should return a number.
    testString: assert(typeof productSumNumbers() === 'number');
  - text: <code>productSumNumbers()</code> should return 7587457.
    testString: assert.strictEqual(productSumNumbers(), 7587457);

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function productSumNumbers() {

  return true;
}

productSumNumbers();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```

</section>