# 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.