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:
 * $3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, \ldots$

The sequence of those values of $m$ begins:
 * $5, 19, 101, 619, 4421, 35 \, 899, 3 \, 301 \, 819, 1 \, 226 \, 280 \, 710 \, 981, \ldots$

Proof
Let $\map f n$ be defined as:
 * $\map f n := \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$

First we observe that for $n > 1$:
 * $\map f n := n! - \map f {n - 1}$

We have:

From here on in the numbers become unwieldy.