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for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
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Is $\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}$ composite for $n\geq 4$?
Define $a_n$ as follows:
$$
a_1=1,\ \ a_{n+1}=na_n+1\
$$
At this time, the sequence $a_n$ is as follows:
$$
a_n=\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}
$$
I made some discoveries about this sequence.
The …