# 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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