---
id: 5900f45d1000cf542c50ff70
title: 'Problem 241: Perfection Quotients'
challengeType: 5
forumTopicId: 301888
dashedName: problem-241-perfection-quotients
---

# --description--

For a positive integer $n$, let $σ(n)$ be the sum of all divisors of $n$, so e.g. $σ(6) = 1 + 2 + 3 + 6 = 12$.

A perfect number, as you probably know, is a number with $σ(n) = 2n$.

Let us define the perfection quotient of a positive integer as $p(n) = \frac{σ(n)}{n}$.

Find the sum of all positive integers $n ≤ {10}^{18}$ for which $p(n)$ has the form $k + \frac{1}{2}$, where $k$ is an integer.

# --hints--

`perfectionQuotients()` should return `482316491800641150`.

```js
assert.strictEqual(perfectionQuotients(), 482316491800641150);
```

# --seed--

## --seed-contents--

```js
function perfectionQuotients() {

  return true;
}

perfectionQuotients();
```

# --solutions--

```js
// solution required
```