Maximum GCD-Sum of a Subarray

Given an array of positive integers, find a contiguous subarray that maximizes the total GCD-score defined by summing gcd(a[i], a[j]) over all unordered pairs of elements within the subarray. The challenge is to avoid O(n^2) pair checks by using number‑theory optimizations (grouping by divisors, multiplicity counting, prefix techniques) to compute contributions efficiently.