# nt.number theory – Is \$prod_{k=1}^nk^{sigma(k)}\$ a square or a cube for some \$sigmain S_n\$?

Note that for any permutation $$sigmain S_5$$ the product $$prod_{k=1}^5k^{sigma(k)}$$ is neither a square nor a cube.

Question. Let $$n>5$$ be an integer. Is the product $$prod_{k=1}^nk^{sigma(k)}$$ a square for some $$sigmain S_n$$? Is the product $$prod_{k=1}^nk^{sigma(k)}$$ a cube for some $$sigmain S_n$$?

The question looks not very challenging, and I believe that the answer should be positive. Any ideas to provide a proof?

The question can be refined further, for example, I conjecture that for any integer $$n>5$$ there is a permutation $$sigmain S_n$$ with $$sigma(1)=1$$ and $$sigma(2)=2$$ such that $$prod_{k=1}^n k^{sigma(k)}=(p-1)^3$$ for some prime $$p$$. Let $$a(n)$$ denote the number of such permutations $$sigma$$. I find that
$$a(6)=1, a(7)=3, a(8)=2, a(9)=27, a(10)=44, a(11)=154.$$

For the above question, I ask for an actual proof of the positive answer, rather than inaccurate heuristic arguments.

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