The term "left factorial" is sometimes used to refer to the subfactorial 
, whose first few values for 
, 2, ... are 0, 1, 2, 9, 44, 265, 1854, ... (OEIS
A000166).
The same term and notation are also applied, for a nonnegative integer 
, to the factorial
sum
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is the gamma function, 
is the upper incomplete
gamma function on its principal branch, 
is the exponential
integral, and 
is the En-function.
For 
, 1, ..., the first few values are given by 0, 1,
2, 4, 10, 34, 154, 874, ... (OEIS A003422).
The left factorial is always even for 
.
The related factorial sum 
has values 1, 3, 9, 33, 153, 873,
5913, ... for 
, 2, ... (OEIS A007489).
is reported to be prime
or a probable prime for 
, 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354,
721, 1796, 3733, 4769, 9316, 12221, 41532, 78981, 88997, 91743, ... (OEIS A100614;
Mijajlović 2025). The term with 
was found by E. W. Weisstein (Oct. 19,
2006), and S. Batalov reported 
in Jul. 2017. Mijajlović (2025) reported the
three indices 78981, 88997, and 91743 and a search limit of 197,651. These values
have 352,508, 401,827, and 415,436 decimal digits,
respectively. These three values passed strong
pseudoprime tests; certified primality proofs were not reported.
." Math. Balkanica 1, 147-153, 1971.Kurepa,
D. "Left Factorial Function in Complex Domain." Math. Balkanica 3,
297-307, 1973.Mijajlović, Ž. "Kurepa Prime Numbers."
2025.