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Let $e_n^0=n^{-3/4}\log p_3(n)$
Define

(1)
\begin{align} e_n^r := \frac{n^{1/4} e_n^{r-1}-(n-r)^{1/4}e_{n=1}^{r-1}}{n^{1/4}-(n-r)^{1/4}} \end{align}
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