<p>A <b>monic polynomial</b> is a single-variable polynomial in which the coefficient of highest degree is equal to 1.</p>
<p>Define $\mathcal{F}$ to be the set of all monic polynomials with integer coefficients (including the constant polynomial $p(x)=1$). A polynomial $p(x)\in\mathcal{F}$ is <i>cyclogenic</i> if there exists $q(x)\in\mathcal{F}$ and a positive integer $n$ such that $p(x)q(x)=x^n-1$. If $n$ is the smallest such positive integer then $p(x)$ is $n$<i>-cyclogenic</i>.</p>
<p>Define $P_n(x)$ to be the sum of all $n$-cyclogenic polynomials. For example, there exist ten 6-cyclogenic polynomials (which divide $x^6-1$ and no smaller $x^k-1$):</p>
$$\begin{align*}&x^6-1&&x^4+x^3-x-1&&x^3+2x^2+2x+1&&x^2-x+1\\
&x^5+x^4+x^3+x^2+x+1&&x^4-x^3+x-1&&x^3-2x^2+2x-1\\
&x^5-x^4+x^3-x^2+x-1&&x^4+x^2+1&&x^3+1\end{align*}$$<p>giving</p>
$$P_6(x)=x^6+2x^5+3x^4+5x^3+2x^2+5x$$
<p>Also define</p>
$$Q_N(x)=\sum_{n=1}^N P_n(x)$$<p>It's given that
$Q_{10}(x)=x^{10}+3x^9+3x^8+7x^7+8x^6+14x^5+11x^4+18x^3+12x^2+23x$ and $Q_{10}(2) = 5598$.</p>
<p>Find $Q_{10^7}(2)$. Give your answer modulo $1\,000\,000\,007$.</p>