ABC Hits

The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.
We shall define the triplet of positive integers $(a, b, c)$ to be an abc-hit if:
$\gcd(a, b) = \gcd(a, c) = \gcd(b, c) = 1$
$a \lt b$
$a + b = c$
$\operatorname{rad}(abc) \lt c$
For example, $(5, 27, 32)$ is an abc-hit, because:
$\gcd(5, 27) = \gcd(5, 32) = \gcd(27, 32) = 1$
$5 \lt 27$
$5 + 27 = 32$
$\operatorname{rad}(4320) = 30 \lt 32$
It turns out that abc-hits are quite rare and there are only thirty-one abc-hits for $c \lt 1000$, with $\sum c = 12523$.
Find $\sum c$ for $c \lt 120000$.

To solve this problem, we can follow these steps:

1. Create a helper function to compute the radical of a number. This function will take a number as input and return its radical. We can use prime factorization to compute the radical.

2. Implement a method to generate triplets of numbers that satisfy the given conditions. We can start by generating pairs of coprime numbers (a, b) such that a < b. Then, we can compute c = a + b and check if the radical of abc is less than c. 3. Iterate through all possible values of a < b < c < 120,000 to find the abc-hits. 4. Sum up the values of c for all the abc-hits found. Now let's write the Python code to solve the problem: ```python import math def compute_radical(n): # Helper function to compute the radical of a number rad = 1 for i in range(2, int(math.sqrt(n)) + 1): if n % i == 0: rad *= i while n % i == 0: n //= i if n > 1:
rad *= n
return rad

def find_abc_hits(limit):
abc_hits = []
for a in range(1, limit):
for b in range(a + 1, limit – a):
c = a + b
if c >= limit:
break
if math.gcd(a, b) == 1 and math.gcd(a, c) == 1 and math.gcd(b, c) == 1 and compute_radical(a * b * c) < c: abc_hits.append(c) return abc_hits limit = 120000 abc_hits = find_abc_hits(limit) sum_c = sum(abc_hits) print("Sum of c for c < 120,000: ", sum_c) ``` When you run this code, it will output the sum of all `c` values for `c < 120,000` that satisfy the conditions of being an abc-hit. This may take some time to run since the computation involves a large range of numbers, but it should give you the correct answer.

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