---
id: 5900f49b1000cf542c50ffad
title: 'Problem 302: Strong Achilles Numbers'
challengeType: 5
forumTopicId: 301956
dashedName: problem-302-strong-achilles-numbers
---

# --description--

A positive integer $n$ is powerful if $p^2$ is a divisor of $n$ for every prime factor $p$ in $n$.

A positive integer $n$ is a perfect power if $n$ can be expressed as a power of another positive integer.

A positive integer $n$ is an Achilles number if $n$ is powerful but not a perfect power. For example, 864 and 1800 are Achilles numbers: $864 = 2^5 \times 3^3$ and $1800 = 2^3 \times 3^2 \times 5^2$.

We shall call a positive integer $S$ a Strong Achilles number if both $S$ and $φ(S)$ are Achilles numbers. $φ$ denotes Euler's totient function.

For example, 864 is a Strong Achilles number: $φ(864) = 288 = 2^5 \times 3^2$. However, 1800 isn't a Strong Achilles number because: $φ(1800) = 480 = 2^5 \times 3^1 \times 5^1$.

There are 7 Strong Achilles numbers below ${10}^4$ and 656 below ${10}^8$.

How many Strong Achilles numbers are there below ${10}^{18}$?

# --hints--

`strongAchillesNumbers()` should return `1170060`.

```js
assert.strictEqual(strongAchillesNumbers(), 1170060);
```

# --seed--

## --seed-contents--

```js
function strongAchillesNumbers() {

  return true;
}

strongAchillesNumbers();
```

# --solutions--

```js
// solution required
```