68 lines
1.9 KiB
Markdown
68 lines
1.9 KiB
Markdown
---
|
|
id: 5900f4301000cf542c50ff42
|
|
title: 'Problem 196: Prime triplets'
|
|
challengeType: 5
|
|
forumTopicId: 301834
|
|
dashedName: problem-196-prime-triplets
|
|
---
|
|
|
|
# --description--
|
|
|
|
Build a triangle from all positive integers in the following way:
|
|
|
|
$$\begin{array}{rrr}
|
|
& 1 \\\\
|
|
& \color{red}{2} & \color{red}{3} \\\\
|
|
& 4 & \color{red}{5} & 6 \\\\
|
|
& \color{red}{7} & 8 & 9 & 10 \\\\
|
|
& \color{red}{11} & 12 & \color{red}{13} & 14 & 15 \\\\
|
|
& 16 & \color{red}{17} & 18 & \color{red}{19} & 20 & 21 \\\\
|
|
& 22 & \color{red}{23} & 24 & 25 & 26 & 27 & 28 \\\\
|
|
& \color{red}{29} & 30 & \color{red}{31} & 32 & 33 & 34 & 35 & 36 \\\\
|
|
& \color{red}{37} & 38 & 39 & 40 & \color{red}{41} & 42 & \color{red}{43} & 44 & 45 \\\\
|
|
& 46 & \color{red}{47} & 48 & 49 & 50 & 51 & 52 & \color{red}{53} & 54 & 55 \\\\
|
|
& 56 & 57 & 58 & \color{red}{59} & 60 & \color{red}{61} & 62 & 63 & 64 & 65 & 66 \\\\
|
|
& \cdots
|
|
\end{array}$$
|
|
|
|
Each positive integer has up to eight neighbours in the triangle.
|
|
|
|
A set of three primes is called a prime triplet if one of the three primes has the other two as neighbours in the triangle.
|
|
|
|
For example, in the second row, the prime numbers 2 and 3 are elements of some prime triplet.
|
|
|
|
If row 8 is considered, it contains two primes which are elements of some prime triplet, i.e. 29 and 31. If row 9 is considered, it contains only one prime which is an element of some prime triplet: 37.
|
|
|
|
Define $S(n)$ as the sum of the primes in row $n$ which are elements of any prime triplet. Then $S(8) = 60$ and $S(9) = 37$.
|
|
|
|
You are given that $S(10000) = 950007619$.
|
|
|
|
Find $S(5678027) + S(7208785)$.
|
|
|
|
# --hints--
|
|
|
|
`primeTriplets()` should return `322303240771079940`.
|
|
|
|
```js
|
|
assert.strictEqual(primeTriplets(), 322303240771079940);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function primeTriplets() {
|
|
|
|
return true;
|
|
}
|
|
|
|
primeTriplets();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|