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We seek the largest integer that divides $P$ for all odd $n$. Since $n$ is odd, all terms are odd, so $P$ is odd — 2 does not divide it.
We analyze divisibility by small primes:
Divisibility by 3**: Among any four consecutive odd integers, at least one is divisible by 3. This can be shown by checking modulo 3: the residues of odd integers mod 3 are 0, 1, or 2. In any block of 6 consecutive integers, four odd ones must include one divisible by 3. So $3 \mid P$.

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