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Triangle Triples

Problem 378

Published on Sunday, 1st April 2012, 07:00 am; Solved by 403

Let T(n) be the nth triangle number, so T(n) =
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 ≤ 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.