Theorem: Sum of Sequence of Alternating Positive and Negative Factorials being Prime

Let $n \in \Z_{\ge 0}$ be a positive integer.
Let:
{{begin-eqn}}
{{eqn | l = m
      | r = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!
      | c = 
}}
{{eqn | r = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1
      | c = 
}}
{{end-eqn}}
The sequence of $n$ such that $m$ is prime begins:
:$3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, \ldots$
{{OEIS|A001272}}
The sequence of those values of $m$ begins:
:$5, 19, 101, 619, 4421, 35 \, 899, 3 \, 301 \, 819, 1 \, 226 \, 280 \, 710 \, 981, \ldots$
{{OEIS|A071828}}

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:
{{begin-eqn}}
{{eqn | l = \map f 1
      | r = 1!
      | c = 
}}
{{eqn | r = 1
      | c = which is not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 2
      | r = 2! - \map f 1
      | c = 
}}
{{eqn | r = 2 - 1
      | c = Examples of Factorials
}}
{{eqn | r = 1
      | c = which is not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 3
      | r = 3! - \map f 2
      | c = 
}}
{{eqn | r = 6 - 1
      | c = Examples of Factorials
}}
{{eqn | r = 5
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 4
      | r = 4! - \map f 3
      | c = 
}}
{{eqn | r = 24 - 5
      | c = Examples of Factorials
}}
{{eqn | r = 19
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 5
      | r = 5! - \map f 4
      | c = 
}}
{{eqn | r = 120 - 19
      | c = Examples of Factorials
}}
{{eqn | r = 101
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 6
      | r = 6! - \map f 5
      | c = 
}}
{{eqn | r = 720 - 101
      | c = Examples of Factorials
}}
{{eqn | r = 619
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 7
      | r = 7! - \map f 6
      | c = 
}}
{{eqn | r = 5040 - 619
      | c = Examples of Factorials
}}
{{eqn | r = 4421
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 8
      | r = 8! - \map f 7
      | c = 
}}
{{eqn | r = 40 \, 320 - 421
      | c = Examples of Factorials
}}
{{eqn | r = 35 \, 899
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f 9
      | r = 9! - \map f 8
      | c = 
}}
{{eqn | r = 362 \, 880 - 35 \, 899
      | c = Examples of Factorials
}}
{{eqn | r = 326 \, 981 = 79 \times 4139
      | c = which is not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f {10}
      | r = 10! - \map f 9
      | c = 
}}
{{eqn | r = 3 \, 628 \, 800 - 326 \, 981
      | c = Examples of Factorials
}}
{{eqn | r = 3 \, 301 \, 819
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f {11}
      | r = 11! - \map f {10}
      | c = 
}}
{{eqn | r = 39 \, 916 \, 800 - 3 \, 301 \, 819
      | c = Factorial of $11$
}}
{{eqn | r = 36 \, 614 \, 981 = 13 \times 2 \, 816 \, 537
      | c = which is not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f {12}
      | r = 12! - \map f {11}
      | c = 
}}
{{eqn | r = 479 \, 001 \, 600 - 36 \, 614 \, 981
      | c = Factorial of $12$
}}
{{eqn | r = 442 \, 386 \, 619 = 29 \times 15 \, 254 \, 711
      | c = which is not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f {13}
      | r = 13! - \map f {12}
      | c = 
}}
{{eqn | r = 6 \, 227 \, 020 \, 800 - 36 \, 614 \, 981
      | c = Factorial of $13$
}}
{{eqn | r = 5 \, 784 \, 634 \, 181 = 47 \times 1427 \times 86 \, 249
      | c = which is not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f {14}
      | r = 14! - \map f {13}
      | c = 
}}
{{eqn | r = 87 \, 178 \, 291 \, 200 - 5 \, 784 \, 634 \, 181
      | c = Factorial of $14$
}}
{{eqn | r = 81 \, 393 \, 657 \, 019 = 23 \times 73 \times 211 \times 229 \, 751
      | c = which is not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map f {15}
      | r = 15! - \map f {14}
      | c = 
}}
{{eqn | r = 1 \, 307 \, 674 \, 368 \, 000 - 81 \, 393 \, 657 \, 019
      | c = Factorial of $15$
}}
{{eqn | r = 1 \, 226 \, 280 \, 710 \, 981
      | c = which is prime
}}
{{end-eqn}}
From here on in the numbers become unwieldy.