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\prod_{k=1}^\infty 1+x^{p_k}=\exp\left(\sum_{q=2}^\infty \frac{1}{q}\sum_{p|q} (-1)^{q+1}p^*x^q\right)  \end{equation}  where $p^*$ means \begin{equation}  p^*=-2, p^*=(-1)^{q/2}2,  p=2\\ p^*=p, \mathrm{otherwise}  \end{equation}