Smarandache ceil functions

In this paper some defmitions, examples and conjectures are exposed related to the Smarandache type functions, found in the Archives of the Arizona State University, Tempe, USA Special Collections. (I) Smarandache Ceil Function of Second Order: 2,4,3,6,4,6,10,12,5,9,14,8,6,20,22,15,12,7,10,26, 18,28,30,21,8,34,12,15,38,20,9,42, 44,30,46,24,14,33,10,52, 18,28,58,39,60, 11,62,25,42, 16,66,45,68,70,12,21,74,30,76,51, 78,40,18,82,84,13,57,86, ... (S (n) = m, where m is the smallest positive integer for which n divides mI\2.) 2 Reference: (a) Surfing on the Ocean of Numbers a few Smarandache Notions and Similar Topics, by Heruy Ibstedt, Erhus University Press, Vail, USA, 1997; p. 27-30. (2) Smarandache Ceil Function of Third Order: 2,2,3,6,4,6,10,6,5,3,14,4,6,10,22,15, 12,7,10,26,6,14,30,21,4,34,6,15,38,20,9,42,22, 30,46,12,14,33,10,26,6,28,58,39,30,11,62,5,42,8,66, 15,34,70, 12,21,74,30,38,51,78,20, 18,82,42,13,57,86, ... (S (n) = m, where m is the smallest positive integer for which n divides mI\3.) 3 Reference: (a) Surfing on the Ocean of Numbers a few Smarandache Notions and Similar Topics, by Heruy Ibstedt, Erhus University Press, Vail, USA, 1997; p. 27-30. (3) Smarandache Ceil Function of Fourth Order: 2,2,3,6,2,6,10,6,5,3,14,4,6,10,22,15,6,7, 10,26,6, 14,30,21,4,34,6, 15,38, 10,3,42,22, 30,46,12, 14,33,10,26,6,14,58,39,30,11,62,5,42,4,66,15,34,70, 6, 21, 74, 30, 38, 51, 78,20, 6,82,42, 13,57,86, ... (S (n) = m, where m is the smallest positive integer for which n divides mI\4.) 4 Reference: (a) Surfing on the Ocean of Numbers a few Smarandache Notions and Similar Topics, by Heruy Ibstedt, Erhus University Press, Vail, USA, 1997; p. 27-30.