How Many Good Permutations?

A permutation \((a_1, a_2, \cdots , a_{2014})\) of \((1, 2, \cdots , 2014)\) is called good if for all positive integers \(k \leq 2014,\) \[2 (a_1 + a_2 + \cdots + a_k)\] is a multiple of \(k.\) Find the last three digits of the number of good permutations of \((1, 2, \cdots , 2014).\)

Details and assumptions

  • The general condition is \[k \mid \displaystyle 2 \displaystyle \sum_{i=1}^{k} a_i \] for all positive integers \(k \leq 2014.\)

  • This problem is not original.

Image credit: Wikipedia Nicolae-boicu
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