
COMMENTS

Sum[ k!, {k,0,n} ] = n! + !n = !(n + 1) = A003422(n+1), where !n is left factorial !n = Sum[ k!, {k,0,n1} ] = A003422(n) = {0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114, 4037914, ...}. Left factorials are even for n>1. Corresponding numbers n such that Sum[ k!, {k,0,n} ]/2 = A003422(n+1)/2 is prime are listed in A124375(n) = {2,3,4,7,8,9,10,29,75,162,270,272,353,...}.


MATHEMATICA

f=0; Do[f=f+n!; If[PrimeQ[f/2], Print[{n, f/2}]], {n, 0, 353}]
