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Problem 127

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abc-hits

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.

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abc匹配

数n的基rad(n)被定义为n的不同质因数之积。例如504 = 2³ × 3² × 7，所以rad(504) = 2 × 3 × 7 = 42。

我们定义正整数三元组(a, b, c)为abc匹配，当其满足如下条件：

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

例如，(5, 27, 32)是一个abc匹配，因为：

1. GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1
2. 5 < 27
3. 5 + 27 = 32
4. rad(4320) = 30 < 32

实际上，abc匹配是非常稀少的，对于c < 1000，只有31组abc匹配，在这些匹配中∑c = 12523。

对于c < 120000，求∑c。

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