Question

Brilli the ant placed the numbers 1,2,…,n in order clockwise around a circle. She can create an infinite sequence A={aj}∞j=0 of integers by letting a0=k∈{1,2,…,n} and constructing ai+1 from ai by taking the integer that is ai positions clockwise in the circle from ai. For a set of sequences S={A0,A1,…,Am}, 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,…,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.