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