I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau necklace polynomials

$$a(p,n)=\frac{n^p-n}{p}.$$

(This is reminiscent of the cyclotomic polynomials, $\Phi_p(x)=\frac{x^p-1}{x-1}$ for the prime indices, and indeed they are related.)

The primes and colored necklaces are both well plumbed but vast waters, so maybe formulas for the non-prime rows are well-known to number theorists / combinatorialists, but it's hard to track them down. Familiar to anyone? Any references?