Problem #68: Is\[\sum_{n\geq 2}\frac{1}{n!-1}\]irrational?

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Progress

The decimal expansion is A331373 in the OEIS. Weisenberg has observed that this sum can also be written as\[\sum_{k\geq 1}\sum_{n\geq 2}\frac{1}{(n!)^k}.\]Erdős [Er88c] notes that $\sum \frac{1}{n!+t}$ should be transcendental for every integer $t$.