is the smallest integer for prime such that is divisible by . If for all , then never divides any sum for all . 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, 20, 7, 57, 6, ... (OEIS A056985).
Ashbacher, C. "Some Properties of the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Math. Informatics Quart.7,
114-116, 1997."Functions in Number Theory." http://www.gallup.unm.edu/~smarandache/FUNCT1.TXT.Mudge,
M. "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."
is the smallest integer for 
prime such that ![Sigma[SW(p)]](/images/equations/Smarandache-WagstaffFunction/Inline3.svg)
is divisible by 
. If 
for all 
, then 
never divides any sum for all 
. 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, 20, 7, 57, 6, ... (OEIS A056985).
