Count Subarrays Whose Product Has an Odd Divisor Count

Given an integer array `arr`, call a nonempty contiguous subarray special when the product of its elements has an odd number of positive divisors. Implement `countSpecialSubarrays(arr)` and return the total number of special subarrays. Subarrays with different starting or ending indices count separately, even when they contain identical values. ### Input and Output - Input: an integer array `arr` of length `n`. - Output: the count of special nonempty contiguous subarrays, using a 64-bit integer type where required. - A positive divisor is a positive integer that divides the product without a remainder. ### Constraints & Assumptions - `1 <= n <= 2 * 10^5` - `1 <= arr[i] <= 200` - All elements are positive; there are no zeros or negative numbers. - The array elements are small, but a subarray's product may be much larger than a machine integer. ### Examples **Example 1** ```text arr = [4, 3, 12] Output: 3 ``` The special subarrays are: - `[4]`: product `4`, with `3` positive divisors. - `[3, 12]`: product `36`, with `9` positive divisors. - `[4, 3, 12]`: product `144`, with `15` positive divisors. The other three subarrays, `[3]`, `[12]`, and `[4, 3]`, are not special. **Example 2** ```text arr = [9, 9, 9] Output: 6 ``` All six nonempty contiguous subarrays are special. Subarrays at different positions count separately. ### Clarifying Questions to Ask - Are identical sequences at different indices counted separately? Yes. - Can elements be zero or negative? No; each is between `1` and `200`. - Can the answer exceed a 32-bit signed integer? Yes, the number of subarrays can be `n * (n + 1) / 2`. ```hint First characterize positive integers with an odd number of divisors. Then consider what information about each prime exponent matters when multiplying array elements. ``` ### What a Strong Answer Covers The answer should justify the number-theoretic characterization, represent products without multiplying them directly, and count qualifying contiguous ranges without enumerating all O(n squared) candidates. It should explain the prefix-state invariant, the empty-prefix initialization, the handling of `1`, and integer-width requirements. ### Follow-up Questions - How would the state representation change if array values could contain more distinct prime factors than fit in one machine word? - Why must the empty prefix be counted before processing the first element? - How could you count qualifying subarrays offline after computing all prefix states?