a(n) = A049433(n)-1 because n!-(n-1)!-1 = (n-1)*(n-1)!-1.

MATHEMATICA

Select[Range[2600], PrimeQ[#(#!)-1]&] (* Because a(22) generates a 7524 digit number to be tested for primality, the program takes a long time to run. *) (* Harvey P. Dale, May 27 2014 *)