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abc-hits

Problem 127

Published on Friday, 1st September 2006, 06:00 pm; Solved by 3097

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:

  1. GCD(a, b) = GCD(a, c) = GCD(b, c) = 1
  2. a < b
  3. a + b = c
  4. rad(abc) < c

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

  1. GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1
  2. 5 < 27
  3. 5 + 27 = 32
  4. 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.