On certain generalizations of the Smarandache function

in [1 J. Our aim is to shmv that certain results from om recent paper [3] can be obtained in a simpler way from a generalization of relation (1). On the other hand, by the method of Le [lJ we can deduce similar, more complicated inequalities of type (1). 2. By mathematical induction we have from (1) immediately: (2) for all integers ai 2 1 (i = L..., n). When al =... = an = n we obtain (3) For three applications of this inequality. remark that S((m!t):::; nS(m!) = nm (4) since S ( m!) = m. This is inequality 3) part 1. from [3J. By the same way, S ( (n!)(n-l)!):::; (n- l)!S(n!) = (n- l)!n = n!, i.e. (5) 125 Inequality (5) has been obtain<~d in [3] by other arguments (see 4) part 1.). Finally, by 5(n 2) ~. 25(n) ~ n for n even (see [3], ineqllality 1), n> 4, we have obtained a refinement of 5 ( n2) ~ n: (6) for n> 4, even. 3. Let m he a divisor of n. i.e. n = km. Then (1) gives 5(n) = 5(km) ~ 5(m) + 5(k), so we obtain: If min, then 5(n)- 5(m) ~ 5 (:). (7) As an application of (7), let d ( n) be the number of divisors of n. Since IT k = n d(n)/2, kin and IT k = n1 (see [3]), and by IT /.:1 IT k, from (7) we can deduce that k<n kin k<n (8) This improves our relation (10) from [3]. 4. Let 5(a) = u. 5(b) = v. Then blv1 and u!jx(x-l)... (x-u+l) for all integers x ~ u. But from alu1 we have alx(x- 1)... (x- u + 1) for all x ~ u. Let x = u + v + k (k ~ 1)