We are to thread $n$ beads of $c$'s colors onto a string. The color arrangement of the beads is such as to be symmetric with respect to the center of the string. Yet the color arrangement of beads on the whole string cannot be comprised of a natural number multiple translating copies of that of the first few beads on the string. More specifically, let the color arrangement of the whole string be $(x_i)_{i=1}^n$. There should be no natural numbers $a$ and $b>1$ such that $n=ab$ and $x_{ak+i}=x_i, \forall\,k\in\{1,2,\cdots,b-1\}, i\in\{1,2,\cdots,a\}$. How many distinct color arrangements are there for the beads?
A succinct way to state this problem is to find the cardinality of the set of sequences $\big\{x:=(x_i)_{i=0}^{n-1}\in\{1,2,\cdots,c\}^n\,\big|\, x_i=x_{n-1-i},\,\forall i\in\{0,1,\cdots,n-1\} \bigwedge T^lx\ne x,\forall l\in\mathbf Z-\{0\} \big\}$ where $(Tx)_i=x_{i+1\mod n}$.
If $n=4p$ where $p$ is a natural number, the answer would be $c^{2p}-c^p$.
Best Answer
Let $z_n$ be the number of possible necklaces of length $n$. The number of symmetric necklaces on $c$ colors is $$ c^{\frac{n+1}{2}} \text{ for odd}\ n \\ c^{\frac{n}{2}} \text{ for even}\ n $$ However, this overcounts $z_n$ since it includes necklaces that have the property described ("comprised a natural number multiple translating copies of that of the first few beads on the string"). The number of symmetric necklaces of length $n$ that have this property is $$ \sum_{\substack{\forall a, \text{ } a<n \\ a|n }}z_a $$ where each $z_a$ is the number of possible necklaces on $a$ beads.
Note that $z_1=c$. Then the number of possible necklaces is $$ z_n=\begin{cases} c^{\frac{n+1}{2}}-\left(\sum_{\substack{\forall a, \text{ } a<n \\ a|n }}z_a\right) \text{ for odd}\ n \\ c^{\frac{n}{2}}-\left(\sum_{\substack{\forall a, \text{ } a<n \\ a|n }}z_a\right) \text{ for even}\ n \end{cases} $$ The first few values of $z_n$ are $$ z_1=c \\ z_2=0 \\ z_3=c^2-c \\ z_4=c^2-c \\ z_5=c^3-c \\ z_6=c^3-c^2 \\ z_7=c^4-c \\ z_8=c^4-c^2 \\ \vdots $$