Problem 127: abc-hits

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The radical of n, rad(n), is the product of distinct prime factors of n. For example, 504 = 23 × 32 × 7, so rad(504) = 2 × 3 × 7 = 42.

We shall define the triplet of positive integers (a, b, c) to be an abc-hit if:

GCD(a, b) = GCD(a, c) = GCD(b, c) = 1

a < b

a + b = c

rad(abc) < c

For example, (5, 27, 32) is an abc-hit, because:

GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1

5 < 27

5 + 27 = 32

rad(4320) = 30 < 32

It turns out that abc-hits are quite rare and there are only thirty-one abc-hits for c < 1000, with ∑c = 12523.

Find ∑c for c < 120000.