I've decided to extend

this OEIS sequence, which is also known as an

Alternating Factorial. I wrote a custom sieve and a pfgw script to process the output file from the sieve. I intend to sieve and test to n = 100000. Like factorials, this form removes a smaller percentage of candidates than other forms. I have sieved to 4e10 and 44% of the original terms still remain. Sieving at this time has a removal rate of about one-fifth what it needs to be in order to sieve to an appropriate depth.

My program can be easily modified to support

this sequence, also known as a

Factorial Sum. If anyone is interested in taking on such a search, please let me know and I'll cook up some software for you.