51 lines
1.1 KiB
Markdown
51 lines
1.1 KiB
Markdown
---
|
|
id: 5900f53d1000cf542c51004f
|
|
title: 'Problem 464: Möbius function and intervals'
|
|
challengeType: 5
|
|
forumTopicId: 302139
|
|
dashedName: problem-464-mbius-function-and-intervals
|
|
---
|
|
|
|
# --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).
|
|
|
|
# --hints--
|
|
|
|
`euler464()` should return 198775297232878.
|
|
|
|
```js
|
|
assert.strictEqual(euler464(), 198775297232878);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function euler464() {
|
|
|
|
return true;
|
|
}
|
|
|
|
euler464();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|