One consideration is whether there is a one-to-one mapping, and a well defined inverse transform. Then we could consider if a recursive sum exists that has a Laurent series \[p(x) = \frac{2}{x} - \frac{3}{x^3} + \frac{5}{x^5} - \frac{7}{x^7} + \frac{11}{x^9} - \cdots\\ p(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1}p_k}{x^{(2k-1)}}\] a clear candidate to try would be the inverse of the candidate prime sequence above. That is \[p(x)?=\frac{2}{x+\frac{6}{x+\frac{48}{x+\frac{594}{x+\frac{10520}{\cdots}}}}}\]