A polynomial from character theory

Question

Let $G$ be a finite group, and define $a(n)=\#\{g\in G\mid o(g)=n\}$.

In problem 5.18 of Isaacs' Character Theory of Finite Groups, the following polynomial is defined:

$$F_G(X)=\frac{1}{|G|}\sum_ma(m)X^{|G|/m}$$

Theorem: $F_G(k)$ is an integer for $k\in\mathbb{Z}$.

Setting $G$ to be the cyclic group with $n$ elements, define:

$$g_n(X)=\frac{1}{n}\sum_{d|n}\varphi(d)X^{n/d}$$

Questions:

1. Does $g_n$ have some other significance, in some other context? Perhaps in number theory?

2. Is there a more direct way to show that $g_n(k)$ is an integer for every integer $k$?

In the special case that $n=p$ for some prime $p$,

$$g_p(X)=\frac{1}{p}(X^p+(p-1)X)$$.

It follows easily from Fermat's Little Theorem that $g_p(k)$ is an integer for integer values of $k$.

Answer 1

1. It plays a role in combinatorics. $g_n(k)$ counts the number of necklaces with $n$ beads chosen from $k$ colors (up to rotationary symmetry). See https://en.wikipedia.org/wiki/Necklace_(combinatorics)
2. Follows from 1. :-)