49 lines
981 B
Markdown
49 lines
981 B
Markdown
|
---
|
||
|
|
id: 5900f4421000cf542c50ff55
|
||
|
|
title: 'Problem 214: Totient Chains'
|
||
|
|
challengeType: 5
|
||
|
|
forumTopicId: 301856
|
||
|
|
dashedName: problem-214-totient-chains
|
||
|
|
---
|
||
|
|
|
||
|
|
# --description--
|
||
|
|
|
||
|
|
Let φ be Euler's totient function, i.e. for a natural number n,
|
||
|
|
|
||
|
|
φ(n) is the number of k, 1 ≤ k ≤ n, for which gcd(k,n) = 1.
|
||
|
|
|
||
|
|
By iterating φ, each positive integer generates a decreasing chain of numbers ending in 1. E.g. if we start with 5 the sequence 5,4,2,1 is generated. Here is a listing of all chains with length 4:
|
||
|
|
|
||
|
|
5,4,2,1 7,6,2,1 8,4,2,1 9,6,2,1 10,4,2,1 12,4,2,1 14,6,2,1 18,6,2,1
|
||
|
|
|
||
|
|
Only two of these chains start with a prime, their sum is 12.
|
||
|
|
|
||
|
|
What is the sum of all primes less than 40000000 which generate a chain of length 25?
|
||
|
|
|
||
|
|
# --hints--
|
||
|
|
|
||
|
|
`euler214()` should return 1677366278943.
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.strictEqual(euler214(), 1677366278943);
|
||
|
|
```
|
||
|
|
|
||
|
|
# --seed--
|
||
|
|
|
||
|
|
## --seed-contents--
|
||
|
|
|
||
|
|
```js
|
||
|
|
function euler214() {
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
euler214();
|
||
|
|
```
|
||
|
|
|
||
|
|
# --solutions--
|
||
|
|
|
||
|
|
```js
|
||
|
|
// solution required
|
||
|
|
```
|