Triangle Triples
Problem 378
Published on Sunday, 1st April 2012, 07:00 am; Solved by 424; Difficulty rating: 35%
Let T(n) be the n^{th} triangle number, so T(n) = 

. 
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 ≤ i < j < k ≤ n and dT(i) > dT(j) > 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.