2018-09-30 22:01:58 +00:00
---
id: 5900f53d1000cf542c51004f
title: 'Problem 464: Möbius function and intervals'
2020-11-27 18:02:05 +00:00
challengeType: 5
2019-08-05 16:17:33 +00:00
forumTopicId: 302139
2018-09-30 22:01:58 +00:00
---
2020-11-27 18:02:05 +00:00
# --description--
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
The Möbius function, denoted μ(n), is defined as:
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
μ(n) = (-1)ω(n) if n is squarefree (where ω(n) is the number of distinct prime factors of n)
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
μ(n) = 0 if n is not squarefree.
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
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.
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
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).
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
For example, C(10) = 13, C(500) = 16676 and C(10 000) = 20155319.
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
Find C(20 000 000).
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
# --hints--
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
`euler464()` should return 198775297232878.
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
```js
assert.strictEqual(euler464(), 198775297232878);
2018-09-30 22:01:58 +00:00
```
2020-11-27 18:02:05 +00:00
# --seed--
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
## --seed-contents--
2018-09-30 22:01:58 +00:00
```js
function euler464() {
2020-09-15 16:57:40 +00:00
2018-09-30 22:01:58 +00:00
return true;
}
euler464();
```
2020-11-27 18:02:05 +00:00
# --solutions--
2018-09-30 22:01:58 +00:00
```js
// solution required
```