Problem 127
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:
- 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.
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:
- GCD(a, b) = GCD(a, c) = GCD(b, c) = 1
- a < b
- a + b = c
- rad(abc) < c
例如,(5, 27, 32)是一个abc匹配,因为:
- GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1
- 5 < 27
- 5 + 27 = 32
- 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。