Special permutations

We call a positive integer \( n \) a beautiful number if there exists a permutation \( a_1, ..., a_n \) of \( 1,..., n\) such that \( \{a_1-1, a_2-2, a_3-3, ..., a_n-n\} \) and \( \{a_1+1, ..., a_n+n\} \) are both equivalent to \( \{ 1, 2, ..., n \} \) modulo \( n \). How many beautiful numbers are there between 1 and 1000 (inclusive)?

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