(;) Semicolon Math: Sequence Ordering

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Sequence Ordering (ID: 6)

Problem status: not solved

Define a "cyclic swap" as taking a certain string of numbers, \(a_1, a_2, a_3, \dots a_n\) and then moving the last number to the front, thus becoming \(a_n, a_1, a_2, \dots a_{n-1}\). Daniel takes a list of \(n\) integers from \(1,2,3, \dots n\) where \(n>3\) and performs a cyclic swap for each pair of integers in form \(2i - 1\) and \(2i\) where \(i\) is an integer and \(1 \le 2i - 1 \le n\). Then, he performs a cyclic swap for each triple of integers in form \(3i - 2, 3i - 1,\) and \(3i\) where \(i\) is an integer and \(1\le 3i - 2, 3i - 1, 3i \le n\). He also performs a cyclic swap on the remaining \(n \pmod 3\) integers that he hasn't applied the cyclic swap onto. In general, Daniel performs cyclic swaps for each \(k\)-tuple of integers \((ki - k + 1, ki - k + 2, ki - k + 3, \dots, ki)\) where \(k\) and \(i\) are integers with \(2 \le k \le n\) and \(1 \le ki - k + 1, ki \le n\). He then performs a cyclic swap on the remaining \(n \pmod k\) integers that he hasn't applied the cyclic swap onto.

For example, on \(1,2,3,4,5\) the list would go:

The smallest \(n\) in which a number in it's original position on the list is at the same position by the end of the process is \(n=6\). What is the third smallest \(n\)?

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