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 form for $D>6$.

According to sage $j(2)= \,_1F_4\left(\begin{matrix} 1 \\ -i + 1,i + 1,-i \, \sqrt{2} + 1,i \, \sqrt{2} + 1 \end{matrix} ; 1 \right)$

The length of the latex sage's closed form of $j(6)$ is 2534 characters long.

Sage's results in machine readable form are available

Wolfram Alpha can't solve $D>1$ and chatGPT "believes" there isn't simple closed form for $j(2)$.

Q1 Are sage's results correct?

Q2 Is there closed form for $D>6$?