74 lines
1.3 KiB
Markdown
74 lines
1.3 KiB
Markdown
---
|
||
id: 5900f53d1000cf542c51004f
|
||
challengeType: 5
|
||
title: 'Problem 464: Möbius function and intervals'
|
||
---
|
||
|
||
## Description
|
||
<section id='description'>
|
||
The Möbius function, denoted μ(n), is defined as:
|
||
μ(n) = (-1)ω(n) if n is squarefree (where ω(n) is the number of distinct prime factors of n)
|
||
μ(n) = 0 if n is not squarefree.
|
||
|
||
Let P(a,b) be the number of integers n in the interval [a,b] such that μ(n) = 1.
|
||
Let N(a,b) be the number of integers n in the interval [a,b] such that μ(n) = -1.
|
||
For example, P(2,10) = 2 and N(2,10) = 4.
|
||
|
||
|
||
|
||
Let C(n) be the number of integer pairs (a,b) such that:
|
||
1 ≤ a ≤ b ≤ n,
|
||
99·N(a,b) ≤ 100·P(a,b), and
|
||
99·P(a,b) ≤ 100·N(a,b).
|
||
|
||
For example, C(10) = 13, C(500) = 16676 and C(10 000) = 20155319.
|
||
|
||
|
||
|
||
Find C(20 000 000).
|
||
</section>
|
||
|
||
## Instructions
|
||
<section id='instructions'>
|
||
|
||
</section>
|
||
|
||
## Tests
|
||
<section id='tests'>
|
||
|
||
```yml
|
||
- text: <code>euler464()</code> should return 198775297232878.
|
||
testString: 'assert.strictEqual(euler464(), 198775297232878, "<code>euler464()</code> should return 198775297232878.");'
|
||
|
||
```
|
||
|
||
</section>
|
||
|
||
## Challenge Seed
|
||
<section id='challengeSeed'>
|
||
|
||
<div id='js-seed'>
|
||
|
||
```js
|
||
function euler464() {
|
||
// Good luck!
|
||
return true;
|
||
}
|
||
|
||
euler464();
|
||
```
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</section>
|
||
|
||
## Solution
|
||
<section id='solution'>
|
||
|
||
```js
|
||
// solution required
|
||
```
|
||
</section>
|