Doing some related research (more info in a past question), I've stumbled upon this interesting problem which I cannot seem to solve myself:
Let an integer $N$ be the number of digits imprinted on a bracelet, which can come in two values: 1 and 0. Due to the rotational symmetry of a bracelet, every bracelet has $N$ possible "rotations", or sequences of 1's and 0's, associated with it. For example, the sequences "$s_1 = 10010$", "$s_2 = 01001$", "$s_3 = 10100$", etc. are all rotations of the same bracelet (shifting all digits to the right and looping at the end).
The value $V$ of a particular sequence $s$ is decided with the following procedure:
1) The value of a sequence is the sum of the value $v_i$ of its digits $s_i$, where $i$ ranges from 1 (right-most) to $N$, (left-most).
2) The value of the right-most digit $v_1$ is 1 if $s_1 = 1$, or 0 if $s_1$ = 0.
3) The value $v_i$ for $i > 1$ is $v_i = 2^{i-1}*s_i*3^d$, where $d$ is equal to the sum of all digits $s_j$, $j<i$.
As an example, let's take the sequence $s$ = 011. $v_1 = 1$, and $v_2 = 2*1*3 = 6$, and $v_3 = 4*0*9$, and so $V = 7$
As we have seen before, each bracelet has $N$ associated sequences, and each of those has a value associated with it. For our previous example ("011"), we have the other sequences "101", "110". Their associated values are $13$ and $14$ respectively.
To recap, our unique bracelet has the values 7, 13 and 14 associated with its sequences under rotation.
Theorem: For $N$>3 and excluding sequences which repeat before looping $N$ times (such as "000", "1010"), at least two values of every bracelet are relatively co-prime, EDIT: or one of the values is a prime number.
Can you prove or disprove this theorem? In our example, 7 and 13 are co-prime, since they are prime numbers. More examples: The bracelet "00101" has the following values associated to its sequence and rotations: EDIT 13,50,25,52,26.
I think that "1000110" has the sequence $590,295,625,680,340,170,85$, which are all divisible by $5$.