Circle Cover Numbers

Number Theory Level 5

Brilli the ant placed the numbers \(1,2, \ldots, n\) in order clockwise around a circle. She can create an infinite sequence \(A = \{a_j\}_{j=0}^{\infty}\) of integers by letting \(a_0 = k \in \{1,2, \ldots, n\}\) and constructing \(a_{i+1}\) from \(a_{i}\) by taking the integer that is \(a_i\) positions clockwise in the circle from \(a_i\). For a set of sequences \(S = \{A_0, A_1, \ldots, A_m\},\) we say that \(S\) covers the circle if each number on the circle occurs an infinite number of times in the sequences of \(S.\)

We define the Circle Cover Number of \(n\) to be the minimum size of a set \(S\) that covers the circle with entries \(\{1,2, \ldots, n\}\). Determine how many integers from \(1\) to \(100\) have Circle Cover Number equal to \(2\).

Details and assumptions

It is possible for the Circle Cover Number of \(n\) to be infinite, if there is no finite set \(S\) which covers the circle.


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