Problem of the Week 1197

Triplets

Call an integer n a triplet if n = a + b + c, where a, b, c are positive integers such that a < b < c and a divides b divides c.

For example, 7 is a triplet by 1 + 2 + 4 = 7. And it is easy to see that 1, 2, 3, 4, 5, and 6 are not triplets.

How many other nontriplets are there?

Source: 2011 India National Olympiad, in latest issue of Crux. Math.