CN-SMO 2006 Problem 8

Given a circle with its perimeter equal to $n$ ( $n \in N^*$ ), the least positive integer $P_n$ which satisfies the following condition is called the "number of the partitioned circle": there are $P_n$ points ( $A_1,A_2, \ldots ,A_{P_n}$ ) on the circle; For any integer $m$ ( $1\le m\le n-1$ ), there always exist two points $A_i,A_j$ ( $1\le i,j\le P_n$ ), such that the length of arc $A_iA_j$ is equal to $m$ . Furthermore, all arcs between every two adjacent points $A_i,A_{i+1}$ ( $1\le i\le P_n$ , $A_{p_n+1}=A_1$ ) form a sequence $T_n=(a_1,a_2,,,a_{p_n})$ called the "sequence of the partitioned circle". For example when $n=13$ , the number of the partitioned circle $P_{13}$ =4, the sequence of the partitioned circle $T_{13}=(1,3,2,7)$ or $(1,2,6,4)$ . Determine the values of $P_{21}$ and $P_{31}$ , and find a possible solution of $T_{21}$ and $T_{31}$ respectively.