Count all valid formed pairs and sum the GCD of each pair.
You are given an array of integers. You must form pairs according to the problem’s pairing rule, then compute the greatest common divisor (GCD) for every formed pair and return the total sum of those GCD values.
The key challenge is that the number of possible pairs can be large, so a direct brute-force enumeration may be too slow. A correct solution usually relies on counting how many numbers are divisible by each possible GCD value and combining those counts carefully.
Write a function that returns the sum of GCDs over all formed pairs.
Input Format
- An integer array
nums. - The exact pairing rule is determined by the platform problem statement; treat the task as summing GCDs over all valid formed pairs in the given array.
Output Format
- Return a single integer: the total sum of GCD values over all formed pairs.
Constraints
- The array may be large enough that an pair enumeration is not ideal.
- Values are positive integers.
- Use integer arithmetic; the final answer may exceed 32-bit range.
Example 1
Input
nums = [1, 2, 3, 4]
Output
7
Explanation
The formed pairs are not fully specified here, so this is an illustrative example of summing GCDs over all unordered pairs: gcd(1,2)=1, gcd(1,3)=1, gcd(1,4)=1, gcd(2,3)=1, gcd(2,4)=2, gcd(3,4)=1. The sum is 7.
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