How Many Good Permutations?

Probability Level 5

A permutation (a1,a2,,a2014)(a_1, a_2, \cdots , a_{2014}) of (1,2,,2014)(1, 2, \cdots , 2014) is called good if for all positive integers k2014,k \leq 2014, 2(a1+a2++ak)2 (a_1 + a_2 + \cdots + a_k) is a multiple of k.k. Find the last three digits of the number of good permutations of (1,2,,2014).(1, 2, \cdots , 2014).

Details and assumptions

  • The general condition is k2i=1kaik \mid \displaystyle 2 \displaystyle \sum_{i=1}^{k} a_i for all positive integers k2014.k \leq 2014.

  • This problem is not original.

Image credit: Wikipedia Nicolae-boicu
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