The Smarandache-Wagstaff function , for prime, is the smallest positive integer, if it exists, for which divides , where the sum-of-factorials
function is
If
for all ,
then no value exists because every term with is divisible by . Therefore, the values do not exist for 2, 5, 7, 13, 19, 31, ... (OEIS A056985).
The function is defined for , 11, 17, 23, 29, 37, 41, 43, 53, 67, 73, 79, 97, ... (OEIS
A056983), with corresponding values 2, 4, 5,
12, 19, 24, 32, 19, 20, 20, 7, 57, 6, ... (OEIS A056984).
--. "Functions in Number Theory." https://fs.unm.edu/FUNCT1.TXT.Ashbacher, C. "Some Properties of the Smarandache-Kurepa and Smarandache-Wagstaff Functions."
Math. Informatics Quart.7, 114-116, 1997.Mudge, M. R.
"Introducing the Smarandache-Kurepa and Smarandache-Wagstaff Functions."
Smarandache Notions J.7, 52-53, 1996.Mudge, M. "Introducing
the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Abstracts of
Papers Presented to the Amer. Math. Soc.17, 583, 1996.Sloane,
N. J. A. Sequences A056983, A056984,
and A056985 in "The On-Line Encyclopedia
of Integer Sequences."
, for 
prime, is the smallest positive integer 
, if it exists, for which 
divides 
, where the sum-of-factorials
function is
for all 
,
then no value exists because every term 
with 
is divisible by 
. Therefore, the values 
do not exist for 2, 5, 7, 13, 19, 31, ... (OEIS A056985).
, 11, 17, 23, 29, 37, 41, 43, 53, 67, 73, 79, 97, ... (OEIS
A056983), with corresponding values 2, 4, 5,
12, 19, 24, 32, 19, 20, 20, 7, 57, 6, ... (OEIS A056984).
