This function is originated from the exiled Romanian professor Florentin Smarandache. It is defined as follows: For any non-null ~tegers n, S(n) is the smallest integer such that (S(n»! is divisible by n. The importance of the notion is that it characterizes a prime number, i. e. : Let p > 4, then: p is prime if and only if S(p) = p. Another properties: If (a,b) = 1, then S(ab) = max ( S(a), S(b) }; and For any non-null integers, S(ab) ~ S(a) S(b) . {All three found and proved by the author in 1979 (see [3], 15, 12- 13, 65).} If n > 1, then S(n) and n have a proper common divisor. {Found and proved by student Prod~nescu in 1993: as a lemma needed to solve the conjecture formulated by the author in 1979 that: the equation S(n) = S(n 1) has no solutions (see [3], 37, and [30]).} Etc.