Budget Selection from a Generated Sequence — Alternative¶
Codex (GPT-6) alternative: the cheapest sorted prefix. Generate the sequence and take all nonpositive values as in the main solution. If the remaining budget is negative, print zero. Otherwise, sort the positive values and take them from smallest to largest until the next value exceeds the remaining budget.
For any fixed count, the smallest values have the minimum total cost: replacing a selected larger value with an omitted smaller one never increases the sum. Thus a feasible sorted prefix attains its length. Once the next prefix is infeasible, no choice of as many or more positive values can be cheaper. This proves the result correct.
Generation and scanning take \(O(n)\) time; sorting takes worst-case \(O(n\log n)\) . Total space is \(O(n)\) . Use 64-bit arithmetic for the recurrence and budget, and 32-bit storage for generated values. The deterministic sorting bound is useful when a complete ordering is also needed or the instance is smaller.
This program computes correct results for every formal recipe, but its full stress measurements exceed the estimated limit. It does not pass the complete timed evaluation. It remains a backup and cross-check; see individual timings. One sorting implementation exceeding the limit does not prove that every \(O(n\log n)\) algorithm must do so.