From test case 2, it seems that you are asking for the closest prime number no less than n. Please clarify this ambiguity in the problem. Otherwise, the answer to test case 2 is 19 (not 23), which is the closest prime number to 20.
Final statement in problem statement disagrees with Case #1 of n=3.
I made a few quick edits.
I suggest to update this problem with big primes.
Make a checkerboard matrix
Specific Element Count
Matrix with different incremental runs
Volume of a box
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