Sum of Sequence of Alternating Positive and Negative Factorials being Prime

Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.

Let:

The sequence of $n$ such that $m$ is prime begins:
 * $3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, \ldots$

Proof
Let $f \left({n}\right)$ be defined as:
 * $f \left({n}\right) := \displaystyle \sum_{k \mathop = 0}^{n - 1} \left({-1}\right)^k \left({n - k}\right)!$

First we observe that for $n > 1$:
 * $f \left({n}\right) := n! - f \left({n - 1}\right)$

We have:

From here on in the numbers become unwieldy.