projecteuler.net

Triangle Triples

 Published on Sunday, 1st April 2012, 07:00 am; Solved by 770;
Difficulty rating: 35%

Problem 378

Let $T(n)$ be the nth triangle number, so $T(n) = \dfrac{n(n + 1)}{2}$.

Let $dT(n)$ be the number of divisors of $T(n)$.
E.g.: $T(7) = 28$ and $dT(7) = 6$.

Let $Tr(n)$ be the number of triples $(i, j, k)$ such that $1 \le i \lt j \lt k \le n$ and $dT(i) \gt dT(j) \gt dT(k)$.
$Tr(20) = 14$, $Tr(100) = 5772$, and $Tr(1000) = 11174776$.

Find $Tr(60 000 000)$.
Give the last 18 digits of your answer.