Some Elementary Algebraic Considerations Inspired by Smarandache Type Functions ( II )

The paper presents new properties for some functions constructed similarly to the functiol} ry :" N* -N* , the Smarandache function, defined by "In E N*, ry(n) = min{kJk! is divisible by n} "Smarandache's type function" " The Smarandache "I function and its principal properties are already known in the literature of speciality. Other functions were built analogously, among which the following ones. The function TJI. Starting from a sequence of positive integers a : N* -N* satisfying the condition "In E N*, 3mn E N*, "1m ~ mn =:} n/a(m) (1) an associated function was built "II : N* -N* , defined by TJI(n) = min {mnlmn is given by (1) }, "In E N* . (2) Such sequences possibly satisfying an extra condition considered by G. Christol to generalise the p-adic numbers were called also multiplicative convergent to zero (m.c.z.). An example is a : N* -N* with a(n) = n!. For n = 6, there is m~ = 4 such that "1m ~ 4 =:} 6/m! ( 6/4! for m = 4; 6/5! for m = 5) but there is and m~ = 7 such that "1m ~ 7 =:} 6/m!; because the smallest of them is ms = 3 such that "1m ~ 3 =:} 6/3!, it results TJI(6) = 3.