For each integer from zero through n, return its number of set bits.
A good solution should
- State the invariant or decision rule that makes the approach correct.
- Explain time and space costs in terms of input size and required output.
- Handle empty, minimal, duplicate, and boundary-shaped inputs.
Reasoning prompt
Use a recurrence from the lowest set bit or from the value’s half instead of recounting each number independently.
Use the linked judge for the canonical challenge. The examples above are compact TRT checks; create additional tests around the boundary most likely to break your invariant.