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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)^?px^q\right)  \end{equation}  where the pattern for the negative signs is yet to be found. It would appear that if the power of $x$ is divisible by $2$ and the power of $x$ divided by $2$ is not equal to $1$, then the overall sign of the temr in the expansion is negative. Then the polynomials in the numerator must be investigated.