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For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$. For $D \le 6$, sage finds closed form in terms of hypergeometric functions at algrebraic arguments and fails to find closed ...

Working with precision 500 decimal digits, mpmath in sage computes: $$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$ We believe the LHS of \eqref{1} ...