Smarandache-Wagstaff Function

Given the sum-of-factorials function

 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).

REFERENCES:

Ashbacher, C. "Some Properties of the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Math. Informatics Quart. 7, 114-116, 1997.

"Functions in Number Theory." https://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."