When k-to-the-k is a square

For how many integers $k$ with $100 \le k \le 400$ is $k^{k}$ a perfect square?

Show hints (2)+

$k^k=\prod p_i^{a_i k}$ is a square iff every $a_i k$ is even: true if $k$ is even, or ( $k$ odd and) $k$ is a perfect square.
Count evens in $[100,400]$ plus the odd perfect squares; don't double-count even squares.

Answer

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156

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