2018-09-30 22:01:58 +00:00
---
id: 5900f4e61000cf542c50fff9
title: 'Problem 378: Triangle Triples'
2020-11-27 18:02:05 +00:00
challengeType: 5
2019-08-05 16:17:33 +00:00
forumTopicId: 302040
2021-01-13 02:31:00 +00:00
dashedName: problem-378-triangle-triples
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
2021-07-29 19:48:17 +00:00
Let $T(n)$ be the $n^{\text{th}}$ triangle number, so $T(n) = \frac{n(n + 1)}{2}$.
2018-09-30 22:01:58 +00:00
2021-07-29 19:48:17 +00:00
Let $dT(n)$ be the number of divisors of $T(n)$. E.g.: $T(7) = 28$ and $dT(7) = 6$.
2018-09-30 22:01:58 +00:00
2021-07-29 19:48:17 +00:00
Let $Tr(n)$ be the number of triples ($i$, $j$, $k$) such that $1 ≤ i < j < k ≤ n$ and $dT(i) > dT(j) > dT(k)$. $Tr(20) = 14$, $Tr(100) = 5\\,772$ and $Tr(1000) = 11\\,174\\,776$.
2018-09-30 22:01:58 +00:00
2021-07-29 19:48:17 +00:00
Find $Tr(60\\,000\\,000)$. Give the last 18 digits of your answer.
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
2021-07-29 19:48:17 +00:00
`triangleTriples()` should return `147534623725724700` .
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
```js
2021-07-29 19:48:17 +00:00
assert.strictEqual(triangleTriples(), 147534623725724700);
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
2021-07-29 19:48:17 +00:00
function triangleTriples() {
2020-09-15 16:57:40 +00:00
2018-09-30 22:01:58 +00:00
return true;
}
2021-07-29 19:48:17 +00:00
triangleTriples();
2018-09-30 22:01:58 +00:00
```
2020-11-27 18:02:05 +00:00
# --solutions--
2018-09-30 22:01:58 +00:00
```js
// solution required
```